An OTTB at my barn was like this. She would pull the bit to the right side of her mouth. She has teeth problems from poor dentistry when racing and they messed up her jaw. She has TMJ and this also causes her to get pain in her head. Definitely get the dentist out to check, preferably one who doesn’t have to use a bunch of unnecessary equipment, which can get in the way of finding out what the real problem is. My dentist doesn’t use any drugs or sedate the horses, he also doesn’t use anything to keep their mouth open. All of the horses love him and are much better for him than any other dentist.