Lady Mustang, don't take any offense to what has been said. Everyone here truly has your best interest and the horse's best interest in mind.
It's great that you seem to be having fun while you're riding but it really sounds like your instructor is doing things backward and skipping over some of the pretty important basics. From an instructor/trainer point of view; if you can't even trot around without grabbing the horn to keep your balance, then you have absolutely no
business riding on trails where the horses need to gallop up hills...which, IMHO, is a very bad habit to get into for both rider and horse, but that's a whole other thread and debate.
While there is nothing wrong with grabbing the horn when you truly feel off balance, it is really easy to get into the habit of grabbing it all the time for everything. When you're a beginner and you are really off balance, grabbing the horn can help keep you from falling off, but it also has a cost. Grabbing the horn and hanging on tends to make a person stiffen up...which causes their balance to suffer
because their body stops being fluid with the motion of the horse...which results in a floppy sack of potatoes in the saddle.
If you're just wanting to ride for fun and doing it for free, then it's to be expected that the person riding with you won't actually be teaching you the things you need to know, but if you're paying
to be taught how to properly
ride a horse, then the instructor needs to be teaching
you how to properly ride, not taking you out on trails that are above your ability.