deep breath and RELAX! The way I used to teach my begginers to trot was to take the reins into the outside hand and take hold of the front of the saddle with the other, as you ask the horse to trot pull yourself down deep into the saddle. Make sure that you are sat on your seat bones and not perched forward.

As the horse trots you should be able to feel the motion through your seat and pelvis, try to absorb the movement through your hips and pelvis. You will know when you get it right!! As you start to feel the movement gradually release the downward pull from your inside hand.

Also try riding different horses as some horses are easier to sit to than others and once you have mastered it on one.....its like riding a bike

not read all of the other replies.
However, I think you are describing loss of balance and security in the seat?
At your lessons, can you ask for lunge lessons? These are exceptional for developing your seat, feel and effectiveness of your aids (in this case your legs)
on the lunge lesson you should do some work without stirrups to help deepen your seat, so you dont need your legs to stay on.
It also sounds like you can develop your core strength more (these are the muscles deep in your tummy and lower back- the ones you feel when being sick!!) You can develop this in many ways, by far the best in more time and experience in the saddle. You can also purchase a gym ball to sit on, whilst reading/ watching TV.
If you go on youtube and watch videos of eg Edward Gal you will see that actually hardly any part of his leg lower or upper touches the horse. This is ideal, because when you come to use the leg to send the horse a message (forwards, sideways, up etc) then the horse responds quikcly because your leg has not been touching him the whole time.
Perhaps also see if you can ride in a dressage saddle to help your stability.
Good luck and I hope this helps!

I was going to suggest a lesson with no stirrups.

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Perhaps the work without stirrups should be done in walk first though as starting of without them in trot could make the rider tense up and lose their balance.
In walk try doing a few exercises without stirrups to improve your balance

1/bring knees up and then streach legs down
2/ bring one leg back from the knee then the other then streach legs down
3/round the worlds ! a childs exercise but gives confidence. has to be done while horse is stationary to begin with though This is where the rider does a complete 360 turn bringing one leg over to one side so both legs are together ,then faces back of horse then the other side and finally back to the starting position. Hope I have explained this ok.
4/ standing up in the stirrups bottom off the saddle

These exercises can be done at first keeping one hand holding on to the front of the saddle for balance and once the rider is more balanced and confident they could try some of the exercises not holding on or even in trot.

I am a great believer in hacking out too as sometimes a lesson in a school can be hard as there is a lot of " steering "to think about. On a hack it is more relaxed and therefor the rider can concentrate on enjoying the ride and can think about feeling the movement of the horse without worrying about corners, what their legs and hands are doing etc.

Do you ever get to go out hacking ?

In trot using the hand on the saddle for balance is a good tip , pulling the seat down in to the saddle , relaxing the leg and pelvis and doing a sitting trot. A very good way of feeling the movement , pretend bottom is glued to the saddle
Posting or rising to the trot can be confusing as some riders just think that you count 1 and stand up and 2 sit down but they get the "beat" wrong and that is why they bounce. Dancing to a twist instead of a slow song or vice versa Learning to count out loud to the trot and watching the horses shoulder movement ie sit when the right shoulder comes back can help explain when to sit and when to post/ rise.Also ask the teacher to bandage the horses legs ie off fore and near hind then look at the way the horses legs move in the trot.

Rising by standing on the stirrups is not exactly what a rider should be doing as the movement of rising should actually come from the pelvis and a supple but strong back and it is more of a thrust forward than a standing up movement . Standing in the stirrups can cause a rider to come up too high lose their balance then miss the beat and again this will cause a bounce.

As you can see relaxing, suppleness and balance are very important in riding so as mentioned the exercise ball thing is very good for achieving these things. There is actually a book on riding that tells you how to use the exercise ball to improve all these things but cannot remember the name of the book I shall have a look and see if can find out.

So relax enjoy and learn

I always squeeze on when I am in trot to keep the horse moving forward, if they start to speed up too much i squeeze down the rein or half halt them to slow them down. Sorry if thats wrong but thats the way I was taught!

I was told that if you dont keep your leg on while trotting the horse will slow right down or forget you are on it, you need to be maintaining contact at all times.

I might be talking a load of rubbish but being 50 and coming back to riding after a very long time.......... I find it's all in the hip movement. Even after all this time I find if you let your hips go with the horses movement you don't bounce at all.
I even find sitting trot easier than rising these days. Give it a try.
PatsyB

I am a bit confused.... are we talking about sitting trot or rising trot? lol