I have troubles riding with no stirrups?

Grab a handful of mane or use a neckstrap - my pony is very bouncy too (especially for a 14.1hh pony with quite a long low knee action)

Have a couple of lunge lessons - you won't have to worry about controlling the horse and your instructor can give you some tips on finding your balance

Keep practicing :) It will happen it just takes time. When you ride without stirrups you realise how much you depend on them and stirrups can make a slack rider who falls in the habit of putting their weight in them to rise to the trot instead of using the inner thigh muscles. A horse with a bouncy trot doesn't help but the trot is a bouncy pace after all! I always find the transistion from canter to trot the most difficult especially if you are posting. Try and work on sit trotting bareback as I find that works best for me.

you need to practice your sit trot and make sure its perfect so you can sit the transition it shouldent be that hard your horse is proberly going from a nice working trot and then you ask him to canter and he speeds up his trot before he goes into the cater you are not sitting deep enough and bouncing all over his back or your horse is cantering on the right leg, practice transitions with sturips

Try 'hugging' with your legs. Not to make them forward, but to keep your balance. Then do the downward transition and keep the pressure. This can be hard because sometimes the horse thinks you want them to move forward. Just communicate that you want to trot, and make him do so.

Really? I'm the complete opposite. Fine with no stirrups, a nightmare with. My knees and my ankles get on fire with stirrups.

I would try working on lunging on this horse. Not having to focus on steering too. Get your instructor to lunge the horse around without you having your arms out straight on your sides, or just have them resting on your thighs. If you are still a beginner, however, you may want to put on hand on the pommel or hold some mane with 1 hand.

If you keep contact with your hands to his mouth and also keep your leg on it will make the transition much smoother as your horse will be more balanced, also squeeze your thighs on the saddle like you are trying to keep a dollar bill from slipping away. Good luck :)