I would say that the main conflict in the story is Beauty's mistreatment at the hands of one terrible owner after another. To be fair, Beauty's first couple of owners are wonderful; however, once Squire Gordon is forced to sell Beauty, things turn quite terrible for Beauty. He is mistreated and abused to the point that injury and sickness are the status quo.
Beauty has a momentary respite once Jerry takes ownership of him, but that does not last long. Beauty finds himself once again being worked to near death at the hands of horrible owners, and the weaker he gets, the worse that he is treated.
The resolution begins to come about once Beauty is examined by a farrier. The man says that Beauty is overworked and might actually sell for a decent price if he is allowed decent food and a few days rest. This ends up happening, and Beauty is sold to an old farmer and his grandson. The two characters are wonderful owners and make an effort to rehabilitate Beauty back to the horse that he should be. Their plan is to sell him to a family that will take good care of him. That happens, and Beauty is sold to owners that recognize Beauty as one of Squire Gordon's former prized horses. The story ends with readers knowing that Beauty is back with good horse owners for a long time.