Are you talking about the canter? Or an acutal full out gallop (almost as fast as the horse can go.) If your instructor is asking you to GALLOP and not CANTER when you are not ready find another one. No instructor should push the gallop. Infact, my old English instructor wouldn't even teach it. I've never actually had ANY instructor teach galloping. Next time you go out to the barn to try whatever gait you are scared of listen to the hooves. If they make a 1-2-3 beat it's a canter and if it's a 1-2-3-4 beat it's a gallop. (Walk is also 1-2-3-4 and a trot is 1-2). The one thing I can tell you to do is do lots of trotting work WITHOUT stirrups or even bareback. This will improve your balance amazingly. If you are riding english put a neck strap around the horse and you can hold onto it if you begin to feel yourself slip or you can hold onto their manes...but don't pull to hard. You might need to shorten your stirrups or maybe the feeling of falling is because your saddle isn't tight enough. Good luck!
With tons of practice I can now run barrels, do jumping and galloping bareback. You'll get it sooner or later!