Your fear of falling off isn't helped at all by dwelling on it and it can, in many ways, inhibit yourself from enjoying the ride + learning from it in a positive way. You're so focused on the "what if --" scenario, in the back of your mind if nothing else, that you tense up and do more poorly than you would be if you relaxed.
Riding horses comes with the inherit risk of biting the dust. It doesn't matter if you're riding the most bomb-proofed horse on the calmest day - the risk is still there every time your foot hits the stirrup. It's a risk you've accepted, in every one of your lessons, despite (or perhaps in spite of) your current age. If it helps, sit, breathe and tell yourself that "nothing bad will happen" for a little bit before each lesson. Learn to fully embrace it, because hiding behind the fear of falling only inhibits you. One, from fully enjoying and wringing the experience out of each lesson, and two, from relaxing on your mount and loosening your muscles (which helps you ride better as it is.) Breathe deeply while you ride and don't be afraid to ask for a walking break if you feel like you're getting nervous. They can't force you to trot forever (as much as the sadistic little instructors might like to be able to ;) and on your walk break, breathe some more. Singing does help. A lot.
I might be 19 and telling you this, but when I first fell off of my lease horse I was scared out of my wits to canter him. I almost couldn't get back on him after I was cleared for take off.. I was jittery and in the 5 minutes it took me to come down/be tested I had developed a deep fear and insecurity in myself. I kept thinking to myself that riding wasn't for me, I was inadequate, stupid, all sorts of derogatory stuff, I told myself and resolved that I would quit riding right then and there, untack the horse and go home, never to be seen again. I was a bit over dramatic, but when adrenaline hits me I lose all coherency and I start crying - so whenever I get deeply frightened, angered, or upset, I just cry and beat myself up. (No one said I was perfect.)
It took three months of riding with the "what if I fall" mentality before I was out riding on my own one day, walk/trot (thinking "I'll probably hit the ground, so I shouldn't go any faster than this.) It took the horse spooking, taking off at a full gallop, me losing one stirrup and flopping over on his neck, barreling through two 2-acre arenas and into the grooming area before I learned that thinking "what-if" wouldn't save my hide and that I had to think and act rather than shut down in fear. There was nothing in that field to spook at, and I do firmly believe that being a nervous ninny up on his back fed into the gallop he gave me; I wasn't a strong leader, I was scared of something, and he couldn't find it. So he went into flight mode. I never thought "what-if" before or during a ride again. So don't you get into the habit either. :)
The path is different for you and me, but the journey begins in the heart.