It almost sounds as if you're standing on your stirrups or relying on them more than you should be if losing one makes you fall forward. Your seat shouldn't change when you have lost a stirrup. I'd work on getting it that solid if I was you.
Like others have said it's a combo of balance, position, confidence etc. Practicing riding without stirrups when you want to will be priceless...there's no real way to learn or practice this at home (unless you have horses there), as it's kinda like telling someone how to dance...you've got to be doing it to understand and get the feel.
In english riding much time is spent schooling without stipprups at all gates. When I was a teen I taught myself how to jump bareback before I even learned how to jump with a saddle (not something I would recommend but I thought I was invincible and it seemed natural.) You'll notice show jumpers, even olympic caliber riders lose a stirrup now and again and often have to continue jumping without one if they can't get it back easily.
The idea is to be able to ride at any gait and through whatever you do one horseback without them if need be.
Last edited by EponaLynn; 10-01-2013 at 11:34 PM.