It is unwise to make a blanket recommendation to canter without stirrups. Although it is on the BHS list of required skills, my highy skilled RI warned me not to do this as the downward transition to trot risks throwing new riders off balance.It really is a case of hours in the saddle at canter, and not trying too hard which makes you then brace and bounce more. If you have a safe horse do some canter without stirrups and let your legs relax and hang.
It is unwise to make a blanket recommendation to canter without stirrups. Although it is on the BHS list of required skills, my highy skilled RI warned me not to do this as the downward transition to trot risks throwing new riders off balance.
Every learner has problems with learning canter.
You are not forgetting how to ride canter, but you may be tensing. Are you in a rectangular school with corners? It is easier to learn canter out hacking where you can go some way on a safe bridle track, in a straight line. If you are learning in a dressage school, you can avoid cantering corners by cantering down the long side, trotting the short side and then cantering again on the long side.
There are many people who suggest ways to ride canter but as Carthorse suggests, try doing nothing - breathe deep and slow, sit back, relax and count the 3 beat in your head. My revelation came after hearing Mark Rashid teach. He suggested breathing deep and counting the beat of each pace, 4 for walk. 2 for trot and then 3 for canter.
One problem about learning canter is that it does often (not always) involve more speed than a learner is used to. You have to allow the horse to move freely forward between your legs, relaxing your hands on the reins. If you allow and ask for this free forward motion, then the canter may be less bouncy. Unlike you, it was the short canters I found hardest. A long smooth canter was what I loved.
And a final point is that students learning to canter are rarely given an idea of how far they are expected to canter. Rashid again makes a point of knowing one's destination. Concentrating on cantering to a certain point like a particular tree or a particular corner of the school, conveys the idea to the horse. And when one has done it one can regard it as a success.