I don't know that "natural horsemanship" and "pull the reins as hard as possible" should go together...isn't natural horsemanship about being light and supple?
I'm curious how you found this guy anyway? If he was training your horse, why didn't he let you ride your own horse during the lessons? It seems like that would have been the most logical thing. Either way, it sounds like you need to find someone else to take lessons from.
And I agree with everyone else...in order to become a better rider with better balance and coordination, you have to ride more. Even if all you do is ride in an arena or pasture or wherever for an hour a day at a walk until you feel balanced, and then start trotting little by little until you feel comfortable. Just work with your horse and work on building your confidence.
Also, check out this book. I started working with a horse who hadn't been ridden in a long time, and followed this book and had awesome results! The thing I like most about it is that it has one english rider and one western rider who have problem horses and throughout each exercise, there is an interview with them both that tells you things that were hard for them, and tips to make it work. It's also broken down in a way that is very easy to understand, and there are sections within each exercise "typical problems with the horse & how to correct it" and "typical problems with the rider & how to correct it"
The book starts with groundwork and works up to being in the saddle and all of the exercises teach your horse to be light and supple and it's a great way to build a bond and establish authority with a new horse. I highly recommend!