In a conversation with one of my history professors, the idea that the allies did not have the functional range to carry out targeted bombings was brought up. Was this the case? Or was the concern more about collateral damage to the prisoners?
edit: Thank you all for the answers, this gives me a lot more research to do!
The camps were in range of the bombers (with perhaps a reduced payload for the camps furthest to the east) but the fighters may have struggled to escort them all the way to the target and back (and unescorted American bombers were savaged during daylight raids in '43). Collateral damage would have been a consideration, but a small one. Hitting particular buildings in a camp would have been quite difficult. It was done with medium bombers in Operation Jericho, but that required a low-level approach that was anathema to heavy bombers at the time. Make sure to read this recent answer by /u/estherke. I can give you more information on targeting rail lines below.
The truth is that WWII bombers were largely ineffective against rail lines.
There are a number of reasons for this. The first is that WWII bombers were not accurate enough to reliably hit a rail line. Keep in mind the very nature of a set of rails--they are a ribbon of infrastructure mere feet wide that snakes through the country. Now we add the difficulties of WWII bombing that resulted in most bombs ending up nowhere near their intended target. Specific factories were often missed by the bombers tasked to destroy them, and there were many times in which entire cities were missed by the bombers. The 'solution' to this difficulty was to send more bombers (again, at great expense) so that you could drop more bombs in the hope that one of those bombs would hit its target. This was somewhat effective.
There were operations undertaken against German rail lines. Most prominently, they were targeted as part of the 'Transportation Plan' prior to D-Day. As reconnaissance during the war indicated and surveys after the war confirmed, there was some success for a time. Rail lines, if targeted with large numbers of bombers, could be disrupted--but only for a short time. But unless a bridge or viaduct was somehow heavily damaged or destroyed (something only 'tallboy' and 'grand slam' bombs very late in the war had any real chance of accomplishing), the railroad would be repaired in short order. Basically, most of the time the trains were able to roll through the targeted area in 24 hours. This is partly due to the aforementioned difficulties in targeting rail lines, but the Germans also had dedicated forces to repairing railroad damage. In addition, these units would compel local people to assist in the reconstruction of the rail line. Further, outside of bridges and viaducts railroads are basically piles of rocks and gravel with a little wood and steel beams on top. This means that low-tech solutions were perfectly suitable to fixing the damage caused by relatively high-tech instruments such as heavy four engine bombers equipped with every modern piece of equipment. German railways only collapsed extremely late in the war, basically when every other service had also shut down and the government had effectively collapsed. (Christian Wolmar covers this quite well in Engines of War)
In the final analysis, rail lines could only be cut on a tactical basis. Within a matter of hours repairs on a damaged line could be completed, so to ensure that a rail line was not used it would have to be retargeted multiple times. This meant that the bombers could not be used to attack other targets that were also deemed worthy of their attention. So, to target the railways leading to the death camps, bombers would have to undergo great risks (at great expense both in money and men) repeatedly (because the Germans had great success in repairing rail lines within hours) to even have a chance at success, and other targets would have to be neglected. So the Allies generally focused on other targets rather than try to cut any rail line, much less a rail line leading to any of the camps.
Wolmar also discusses the resilience of German railroads. While there was considerable damage done to German railroads, Wolmar contends that the evidence shows that disruptions to railroad traffic were much less than expected. In particular, he cites a report from civil defense workers sent to Germany after the war to study Germany's railroads. Damage to tracks and controls was usually limited geographically, and tended to not be the subject of repeated attacks in a small area. So the German response was to suspend traffic for a few days and use alternate routes. Apparently, there were many such alternate routes in Germany, though I do not know if there were the same options available near the concentration camps that were mostly in Poland. Also, the Germans had few "triangular junctions" that allowed traffic to move if any one of the three sections were damaged, so their resilience is even more surprising.
Radar is also credited for allowing train traffic to continue despite air raids, as the trains kept rolling until there were planes in the immediate area. Further, trains were not stopped for single aircraft, as the gain was seen as being worth the risk. Also, slave labor and other manpower was used to repair the damage from raids, allowing the lines to be rapidly reopened. Wolmar cites a raid on a railroad yard in Hamm as an example. It was hit by more than 1,300 bombs on April 22, 1944. Six thousand workers were used to get repairs underway, and the trains were rolling through the yard in less than 24 hours. Within six weeks the yard was back to nearly full capacity. If you combine this with the camouflage efforts to simulate damage to operational tracks, you can see that it was difficult to stop the trains.
Wolmar does give some credit to the 1944 campaign to attack the railroads directly. He admits that damage was extensive, but says that the disruptions were more attributable to administrative problems than to the damage itself. He highlights the inability of the Allies to target viaducts and train tunnels until the rather late introduction of the "Tallboy" bombs.
So, attacks on the railroads would have definitely raised the cost of transporting victims to concentration camps, but it may not have been able to bring the trains to a halt. German rail service was remarkably resilient to Allied bombing, and when damage did occur it was often repaired quickly.
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While capability is one reason that military leaders initially gave against bombing the camps, as /u/Domini_canes suggests, a much more important reason has to be considered a political one - leaders argued that the best way to help those in the camps was to defeat the Nazis as quickly as possible. By that logic, diverting resources to bombing the camps or their supply lines was counter-productive. American leaders also worried about propaganda victories by the Germans should Allied Air Forces bomb the prisoners in the process of destroying the camps. There has always been speculation that latent antisemitism was at least partially responsible for this view, but for obvious reasons, there is very little hard evidence that that was the case.
By mid 1944, when the United States had the ability to reach the camps in Poland and Silesia, they argued that it was almost exclusively the diversion of resources that was the problem. This shift was necessary simply because previous US bombing raids had shown that there was, indeed, capability to take out the camps, for certain, if not the railway lines leading to them. The raid on the IG Farben plant at Buna (literally next to Auschwitz III Monowitz and about a half-dozen miles from Birkenau) is an example of just such a bombing.
None-the-less, American leaders refused to divert any resources towards ending the genocide that - to be explicitly clear - American leaders were well aware of in relative detail, including the basic operation of the gas chambers. Exactly why those American leaders decided not to intervene will probably never be known.