The actual content of the movie battles is generally not at all accurate, but yes, there were large-scale battles that occurred in the medieval period with thousands of casualties. Not all of the dead men would have been buried. Some corpses of important nobles might have been identified and taken away by their servants or relatives. Regular soldiers were often buried in mass graves, though. Archaeologists in Portugal found a single grave containing at least 414 individuals from the Battle of Aljubarrota whose bones showed signs of having been left exposed for months before burial. This has generally been taken to mean that this particular grave was for foreign soldiers (either Castilians or French) who died and did not have any local relatives or friends to give them a proper burial. Aljubarrota is just one example of archaeological evidence of medieval battle casualties. Another famous example of mass graves at a battlefield would be the corpses excavated from the Battle of Visby, which took place on the island of Gotland in 1361. The archaeology of that site was extremely valuable, because many of the dead had been buried with their armor on (unlike most medieval mass graves).

Blood Red Roses: The Archaeology of a Mass Grave from the Battle of Towton AD 1461 (2007), edited by Fiorato, Boylston, and Knusel, is about one such mass grave in the UK.