So, based on this, a single turbolaser bolt would have roughly half the power of a hiroshima bomb (~30 TJ). We need 640 TJ to break the Reaper's Kinetic Barriers. Multiply 30TJ by how many turbolaser batteries a Star Destroyer has, and the Star Destroyer still comes out ahead, but on a much smaller scale. The Star Destroyer would need to lay down constant fire from 22 of its batteries to break through the kinetic barriers, with the rest of the blasts doing damage to the Reaper's hull.
Note that the page linked lists 30 TJ as the energy required to melt the asteroid; 250 TJ is the figure given as the estimate for vaporizing the asteroid and the page suggests that vaporization is likely to have occured. Also note that this figure is not for the Star Destroyer's heavy guns, it is for a 'medium-size bolt' presumably from a medium turbolaser cannon. Furthermore, this is for a single bolt. A battery consists of two or more guns unless we're using very nonstandard terminology and so a single battery's fire would consist of two or more bolts per firing cycle.
I would further note that it is not at all clear that the kinetic barriers of the Mass Effect universe would provide any protection at all against turbolaser fire. Lasers are explicitly not blocked by the kinetic barriers of capital ships in Mass Effect, though Reaper shields apparently are partially effective against them (here, last paragraph of the combat section), and I could not say whether or not turbolasers are sufficiently close to lasers in the manner in which they would interact with Mass Effect's kinetic barriers to render Mass Effect's kinetic barriers similarly useless.
I would also question where your figure of 'constant fire from 22 of its batteries' comes from. If the barriers require only 640 TJ to break and each gun fires a bolt which can deliver a minimum of 30 TJ, then that is one bolt from 22 guns (no more than 11 batteries, unless we're using nonstandard terminology; again, in standard nomenclature, a battery consists of two or more guns), not 22 batteries. Furthermore, the kinetic barrier would need to be restored fully between every firing cycle in order to require 'continuous fire' from the Star Destroyer to break the shield, and this page suggests that the turbolasers firing the medium-length turbolasers have the ability to fire one shot every two seconds. Can the Reapers fully restore their shields every two seconds?
Of course, we'd still have to figure out how strong the Star Destroyer's own shields are, because the Reaper might kill the Star Destroyer before it gets a chance to fire all its main batteries.
The only estimates I can think of off the top of my head can be found here and here. The first page suggests that 1e20 Joules is a reasonable estimate of the energy required to break an Imperial Star Destroyer's shields and 3e20 Joules is a reasonable estimate of the kinetic energy required to do the same, though the former value is absorbed over ~30 minutes of engagement and the latter is suggested to be absorbed over the course of 1-2 days. The second page suggests that a shield which was probably already depleted to an unknown degree had an instantaneous impact absorption capacity on the order of 5e14 joules, and sites a book (Anakin Skywalker: The Story of Darth Vader) as stating outright that Star Destroyer shields in peak condition are capable of absorbing multimegaton impacts, though I think the book in question is no longer canonical as of whenever it was that Disney threw out the EU.
If you were to assume that all the weapons on the Star Destroyer fire medium-length bolts each capable of delivering 30 TJ, that each Star Destroyer is capable of firing 5 shots per second (time averaged rate of fire against the Tantive IV in A New Hope, though the Star Destroyer int that engagement may not have been employing all its guns to the fullest extent as the objective of that engagement appears to have been to disable the Tantive IV rather than destroy it), that Star Destroyers and Mon Calamari Cruisers such as used at Endor have comparable firepower, and that each Star Destroyer was engaged by one and only one Mon Calamari Cruiser (which directed all 5 of its 30 TJ shots per second at the destroyer), then in order for the Star Destroyers to be able to engage the Mon Calamari Cruisers for ~30 minutes before shields began failing the Star Destroyer's shields must be capable of absorbing at least 2.7e17 joules over 1800 seconds (30 minutes) before failing. Another estimate of the shield power you could make involves the assumption that the Imperial Star Destroyer is a 'balanced' warship (i.e. a warship whose defenses are capable of resisting its own armament). According to Wookieepedia's Imperial-I Star Destroyer page, an Imperial-I Star Destroyer should carry at least 91 turbolaser cannons. If we assume that 40 of these can be brought to bear for the 'resists its own armament' test (and I would suggest that ~50% of the ship's armament being brought to bear against a single target is something akin to a lower bound on the ability of a Star Destroyer to concentrate its weapons on a single target), then using the 30 TJ per bolt figure an Imperial-I Star Destroyer's shields should be capable of absorbing at least 40 30 TJ bolts impacting roughly simultaneously, or about 1.2e15 joules. If we instead use the 250 TJ figure for vaporizing the asteroid, then the Star Destroyer's shields should have an energy capacity of at least 1.0e16 joules.
As far as how well the Star Destroyer can resist the fire of the Reapers, the main weapon of a capital ship type Reaper is stated to have a yield of 132 to 450 kilotons of TNT, or 5.5e14 to 1.9e15 joules (source). If Star Destroyer shields are as effective against kinetic impactors as they are against turbolasers, then using the lower bound for 30 minutes of combat and assuming that Star Wars shields do not recover over time, an Imperial Star Destroyer's shields should be capable of absorbing 142 to 491 shots from the main weapon of a capital ship type Reaper. Using the 30 TJ per bolt figure for turbolaser bolt energy, a Star Destroyer's shields should be capable of absorbing about 1 to 2 shots from the main weapon of a capital ship type Reaper; the 250 TJ per bolt figure for turbolasers would indicate that the Star Destroyer's shields can absorb about 5 to 18 shots from a capital ship type Reaper's main weapon. I would tend to expect that the Star Destroyer's shield capacity is somewhere between the first estimate and the second if the 30 TJ per bolt figure is to be used as the energy carried by a turbolaser blast regardless of the type of turbolaser used and the likelihood that this is less than the actual amount of energy carried by each medium-length turbolaser bolt.
It would appear from the information provided that if we are generous to the Mass Effect Reapers then the Reapers might be able to take down a Star Destroyer's shields in one shot and then destroy the ship with a follow-on shot. If we are less inclined to go with minimal estimates for Star Destroyer shield capacity, things get worse for the Reapers very quickly.
And yet the entirety of Death Squadron couldn't pump out enough power to overcome that one second-hand shield generator the rebel alliance managed to build on Hoth. Death Squadron had a SSD with 2000 turbolasers and 2000 heavy turbolasers...and at least five Imperial class destroyers as well.
It might be that Death Squadron could do so but that doing so would make it unlikely for anyone to survive the bombardment. Vader's objective at Hoth isn't entirely clear; he wants to destroy the rebel forces, yes, but he also wants to capture Luke and may also want to capture some of the higher-ranking rebels. If the Star Destroyers are capable of a high-accuracy long-range bombardment to destroy the shield generator, hangar bays, and major defensive emplacements, the apparent plan of bombarding the world from outside of the system is not inconsistent with capturing high-value rebels. General Veers' statement that the shield is capable of 'deflecting any bombardment' makes that a questionable interpretation, though it's still possible that there was an implied qualifier on 'any bombardment' of "that allows a reasonable chance of fulfilling the 'capture high-value rebels' objective." General Veers was told to prepare his men for something even before the fleet moved to the Hoth system, Vader is known to be interested in Luke even at that stage of the movie, and later events strongly suggest that the interest is 'capture' rather than 'kill,' which all suggest that the objective of the originally planned orbital bombardment is the disabling of major defenses and the closing off of potential escape routes (destroy hangars, hangar exits, exposed vessels) rather than simple base destruction.
As far as why Rebel warships do not carry shields of the same power, why do you feel that Rebel warships (or, for that matter, Imperial warships) have the necessary power available for that? The Hoth base's power generators are only known to have two major power drains - the theater shield (and much of the base's power should have been redirected to this, according to dialogue in the movie) and the anti-orbital ion cannon. The Hoth base may also have a much larger power source than an average Rebel (or Imperial) warship; Wookieepedia's page indicates that the theater shield was powered by the reactor out of a Praetor-class battlecruiser according to the old expanded universe material, where a Praetor-class battlecruiser was a ship type about 2.5 times the length of an Imperial Star Destroyer and only about ~30 years old at the time of the Battle of Hoth (it was developed in the "final years of the Old Republic," i.e. around the same time that Imperial Star Destroyers began appearing; it may be a design generation or two older, but probably not much more than that). If this is still canonical, or at least representative of the reactor(s) used in the new canon, the base should have had significantly more power available than most of the Mon Calamari Cruisers and Imperial Star Destroyers; Executor and Home One are the main exceptions (along with a couple unidentified potentially large Mon Calamari cruisers which took part in the Endor engagement), and unlike the base those vessels also need to power a significant number of heavy weapons which may be firing at a significant rate for an extended period of time (the base generator(s) powered only a single heavy weapon which fired infrequent bursts of several shots, though it's difficult to assess whether or not this represents a similar, greater, or lesser demand on the available power). The base may also have been able to make use of higher power levels for notionally similar components due to the superior cooling offered by an atmosphere as opposed to vacuum, though at the power levels apparently involved in Star Wars I doubt this would make too much of a difference.
The Reapers have been around for millions of years.
And yet they are not sufficiently advanced relative to the 'new' players on the field to be effectively invincible. The margin of superiority enjoyed by the Reapers is only such that the capital ship type Reapers can engage ~3 capital ships of the species present in the era in which the game is set. These species have only been capable of interstellar spaceflight via mass relay for ~3000 years at the outside, and a species which should have been ~1000 years behind in technology was able to at least compete with the empire with the biggest military. A guy with a club is still a threat to a guy with an assault rifle (though not in most scenarios that aren't fairly contrived), but both of them are very little threat to a guy in a tank, and the Reapers ought to be the guy in the tank with millions of years to develop while the new players on the field are the guys with the club (really recently arrived at interstellar spaceflight) or rifle (older species capable of interstellar spaceflight). They aren't, and while part of that can be attributed to the accelerated development due to the existence of functional relics, your sciences need to be similarly advanced before you can begin to understand even a functioning relic, and your technology and industry needs to be similarly advanced before you can begin making use of anything you learn from that. This means that as far as ancient civilizations go, Prothean and Reaper technology is fairly low-level stuff; Prothean stuff can be understood well enough by species with in-system spaceflight capability to permit those species to reverse-engineer the relics and apply the technology thus developed with little lead time for improving the existing industrial base and overall technology level, and Reaper tech can be understood well enough by species which are so little advanced beyond the initial 'mass effect' discovery that a species which made the discovery within the past couple decades at the outside is militarily competitive with a species that made the discovery a thousand years earlier and which is allied with at least three other species which made the discovery at a similar point or up to ~2000 years before they did that studies of it produce usable results within a few years. Reapers are furthermore insufficiently advanced beyond these that they are vulnerable to the weapons used by these comparative newcomers even without any of the advances gleaned from study of examples of ~modern Reaper technology (as opposed to the presumably older technology represented by the mass relay network, the Citadel, and the stuff that the Reapers do not cleanse during their purge of the galaxy). Some showing for this group that's been around for millions of years when it's facing groups which have had less time to develop by a factor on the order of a thousand or more.