The rider doesn't have steady hands. Just because the hands are still doesn't mean they're steady. Look at the contact when she trots the horse. The rein goes loose TIGHT loose TIGHT and is banging him in the mouth. The horse is curling to try and get away from it.
I bet he is a steady eddie with a rider that actually keeps steady contact. If you work on it, OP, I'm sure this horse will come along very nicely :) Shouldn't be too hard to correct.
"Strength is the ability to use a muscle without tension"