I agree with the above poster. If he can't balance over fences, then he needs to go back to basics.
To me (and please understand I'm not trying to be mean, just saying what I see), it looks like you have more of a balance problem, which could explain why he has a balance problem. By bracing and using his mouth for your own balance, you're solving one problem but creating a whole other issue.. which is that you're causing your horse pain and hindering his jump.
So here is my suggestion: Try free jumping your horse to see how his balance over fences is naturally without any interference. I'm guessing he'll do fine. At that point, you need to start your own balance exercises. Go back to trot poles, cross rails, and cavaletti, and work on these with no reins, no stirrups, etc. Really work on finding your center of balance, and make sure it in no way involves your horse's mouth. Once you have this, I guarantee the rest of your form will improve too.