So you are one fall on your way to becoming a good rider. Falling off is a wonderful lesson that you don't easily forget.
But I think your trainer should have had proper tack for the pony. Sounds like you were more fighting that than actually riding. And if you don't have tack to fit the horse, jump on bareback!! If you want to learn balance, that's the way to do it, because you cannot rely on anything.
Really, a saddle and bridle actually are unnecessary, but they are helpful tools.