I don't think it's "normal" for a green horse to buck. I think if a horse is considered started or broken it should be able to walk, trot, canter, turn and stop without any major issues. If it bucks when it's canters its not properly broken. If it exists in a trained horse then it's a training problem.
On the same note I personally tend to differentiate between instructors and trainers. Instructors teach riders to ride, where as trainers train horses to be ridden. While there is limited crossover I would not expect an instructor to deal with training issues, especially ones such as bucking which can be dangerous, unless they are directly caused by the rider. I've had instructors say to me basically "until your horses behavioural problems are solved, I can not really teach you anything", which is true. An instructor cannot be expected to teach a rider how to control an out of control horse, as that is a skill that takes time and much experience.
If you don't want to engage a trainer, it's probably best to sell your horse, lease that mare for a while until you find something that is really suitable for you.