I'm not usually nervous, but I think I do fall forward. Normally I fall off at the trot and I end up bouncing out.
Easiest way to do that? Warm up at the walk without stirrups. Cross them over your horse's neck and do circles, figures, transition from walk to halt.
Riding bareback is fine too, but it's a different position than being in a saddle.
But both are helpful ways to improve your overall balance. As is working on your core.