From a traditional point of view one would say: “Just make him stop.” How? Well, let’s talk about ways that makes the horse understand. I don’t believe in punishment, because punishment is not telling the horse what you do expect from him, it only tells him to stop the behaviour in the moment.
First step in re-training undesirable behaviour should be: finding the cause of the behaviour. If you know the cause, it is easier to solve it. For instance: if a horse bucks under saddle and the rider punishes the horse by whipping him, it didn’t solve the root of the problem and probably it won’t prevent this behaviour in the future.
Causes can be: pain, fear or the horse has been rewarded for that behaviour in the past. Remember: the receiver (horse) determines if something is rewarding for him.
If the horse bucks under saddle and this behaviour is caused by pain, the horse doesn’t need training to solve the behaviour. Treating the pain will be sufficient in most cases. Check your horse and equipment and get to know his normal behaviour, learn to listen to your horse. It can be caused by poorly fitted saddles, something bothering the horse in the saddle pad, like a splinter, it can be the bridle, the bit that has been put back upside down in the bridle after cleaning, the rider is out of balance, muscle ache, back pain and so on.
Horses can buck because they want to run away from something scary and the rider is trying to prevent it. Here the solution would be to let the horse investigate the fearful thing at his own speed. For the future: practise scary things by doing a lot of de-spooking training.
It can also be a one time scare because another horse was scared or another horse scared him. Anticipate and expose your horse more to these kinds of experiences.
Imagine the horse is ridden by a novice rider or a rider that has been a bit too rough. The horse bucks and the rider is gone. Ping! Rewarded! Or the horse has been bucking in the past, rider fell off and the horse had a chance to go to that juicy patch of grass which he wasn’t allowed to touch by the rider. Ping. Reward. Or the horse bucks and he can run back to his herd once the rider has fallen off.
In these situations make the opposite behaviour more rewarding: change the rough rider into a balanced, light rider who gives lots of rewards, work on the herd bound behaviour.