Is Cantering Scary and also bouncy? I heard it was supposed to be smooth. But is it bouncy the first few times?

I had my first horse when I was eight years old. He was older so the fastest I could go on him was a trot. We had to sell him when we moved. So I started taking riding lessons. Anyways, she made you do basic vaulting first before riding so that you got better balance! Do six tricks at a trot, and then you had to sit let go for a canter for 2 circles ( on a lunge line) As you all probably know you do not have a saddle on vaulting. It is like a pad and then, and then 2 horn type things where you could hang on. My first time cantering I felt like I was going to fall off. I also thought it was very bouncy. Is this normal? It was also on a really little horse not a pony but barely a horse. She also had one hurt leg from a previous injury a few years earlier but she still walked weird. Do you think with a normal sized horse, with no leg injuries and a saddle and a bridle on. It would feel less scary and less bouncy?