An experienced person needs to see this action in person, to decide if you are ready to get on a horse that makes sudden moves.
With a couple years of riding, you have to learn that it is wrong to expect perfection. Sticking on horses when they take off or buck is really an art and it takes a long time for a rider to get that strong, balanced and relaxed.
Don't compare your progress to anyone else. Don't jealously look at others and what they can do. Concentrate on your riding, on yourself and don't worry about anyone else!
If an experienced person says you are not ready for this sort of thing, don't get on that horse. Ride a different one. If there are no appropriate horses for you at that barn, you don't ride there, you go ride somewhere else where they have suitable horses for you to ride. It really is that simple.
Why? It's not just about 'you might get hurt'. If you 'overhorse' yourself, you are going to pick up a LOT of bad habits as you ride. People who 'overhorse' themselves tend to ride too defensively. They are always tight and stiff, and that causes problems their whole riding life - you do NOT want to 'overhorse' yourself and start a lot of bad riding habits.
My friend has two little kids. She gave her daughter a really tough horse for her first horse, 'so she would be tough and a good rider'. It was not broke, it took off, it ran under the arena fence and scraped her off....you name it, this pony did it. This little girl would cry and throw up before her jumping lesson, and this pony was horrible to jump, he would dodge the jump and rear and buck and everything. She tried sooo hard and in the beginning, she LOVED riding.
Mom gave her son, for his first horse, a small pony that was 28 years old and had been everywhere and done everything. All you did was sit up and steer and smile. That little pony would take you anywhere.
So who is the better rider today? Who is jumping, galloping, and winning competitions? The BOY, not the girl. The girl is a nervous wreck who hates competing, and has lots of bad habits and bad form. Hmmm! Guess which is the right way!
Sure, you really want to ride. But if you keep falling off this horse, you should not be riding it.
If anyone can stop it, ever, it's not bolting. A bolter would run through a 4 board wood plank fence, send broken boards flying and keep going.
Why do you fall off?
Oh it could be lots of different things.
Well, you said you freak out. Do you scream and lift your hands up? Or just freeze? Many people freeze. Many people don't even REALIZE that they freeze. They just stop riding, they lean forward and clutch with their hands and legs, and that makes it easier for them to fall off.
When you freeze, even if you just do it the slightest bit, you are stiff - all your muscles and joints are held tight. That means you can't move with your horse when he moves. If you are loose and flexible, you move with your horse.
Also, it's about being 'in the middle of the horse'. If you tend to slide off to one side of the saddle all the time, you will also fall off easily. I would say about 99% of riders actually are not 'in the middle of the horse'. Most of them don't even realize it. They've done it for a long time so it 'feels right' to them. Even if it is a tiny bit, it can make a BIG difference in sticking on during those oh so pleasant moments when horsey bucks, shies or decides to make a quick exit.
All of those 'reasons' and 'why' are just great, but the fact is, if you have ridden for a few years, what is happening to you is NORMAL. Spell it with me now, 'N-O-R-M-A-L'! Yes! You can't expect to be that strong in just such a short time. So don't get mad at yourself or the pony or anything.
Take your riding lessons (not taking lessons???? BAD! Take lessons!), work hard, ride appropriate horses, listen to the instructor and do what she says, and relax and have some fun!
T.T.T. - Hear these three, listen to me, THINGS TAKE TIME, hear them chime'.