If you feel as if and are falling forward...forget grabbing the mane and plant the hands on both sides of the neck for some support.
Doing this will also teach you the "feel" of holding yourself in that position...the horse moving forward in anything faster than a walk actually gives you some impulsion to stay aloft of their back.
"Perching" is a fad trainers now teach.
You need to ride sitting on and around your horse, not perching stiffly above it with your fanny blowing in the breeze.(that is actually how you look, sorry)
That is one of the reasons why you feel insecure...you are "perching" above the horse not riding with that horse between your legs, butt under your torso more and relaxing your back...your back is part of your shock absorber that will absorb the jar and keep you steady and stationary... stiffen and it hurts, it is harder to ride and easier to kiss your butt goodbye on the way to the ground...
Eyes up, heels down, relax the back, let the weight flow and sink into the heels...your ankles should also not be tense as they absorb and truly do move to help your balance...
Me, I think you are being rushed faster & further than you are ready and comfortable doing... all are safety issues.
Tell your trainer you need and want to go back, you want more time to strengthen and get the steps previous to 2-point solidly in place and comfortable...then and only then move on.
If your trainer can't or won't agree with this...maybe a new trainer or approach is needed..you should not be fearful and you sound as if you are.
Good luck...remember, this is supposed to be fun!!