Well, it could certainly be the bit, you'd be surprised by how bad it hurts to be pinched by a snaffle, especially under a rider with heavy hands.
Try it once... have someone hold both rings of a snaffle bit. Wrap your hand snugly around the bit, and ask the person holding it to move the rings like they would if they were riding. It HURTS. Bits with larger rings, egg butts, D rings, O rings, etc aren't as bad for pinching because they allow the rein to slide along the curve of the ring. When you start getting into bits that have a slot or groove punched in the ring to keep the rein stationary... NOT good.
With that said, it could be a number of things. Heavy hands, improperly fitting tack, rider that falls behind the motion, a rider that balances on the bit, the wrong shaped bit for the horses mouth, teeth that need floating (that perhaps have caused abscesses), the horses jaw being out of alignment.... so many things could cause the horse to toss their head. Like spyder says, a video would help.
As for bits, I really like the Robart Pinchless Bits. Very nice, they make a million different versions, but they are designed so that the joint can move freely so it doesn't pinch the horses mouth. You can do the same exercise with a Robart in your palm and have your friend twist away wildly, and you won't get a pinched or bloody palm.
Here is what the Robart joint looks like....