I agree with what Barry said.
I gallop in half-seat, two-point, jumping position, whatever you want to call it (varies by riding discipline and part of the world you live in). There is a reason why eventers and racing jockeys ride in this position. You have far more stability and control than you would sitting up in the saddle. You also are in a better position to follow the horse in your arms and body, because of the way his body moves when he flattens out for the gallop. I also find I can prevent some bucks, as I have pretty good leverage from this position and can make it difficult for the horse to put its head down. If you don't really know how to ride in two-point solidly, take lessons. There is no good way to explain this on the internet.
When I say put leg on I mean squeeze the horse with your calves. Or even thump your leg against him. "Giving him a ton of leg" means squeeze him hard. I'm presuming that at 16 he knows leg on means go forward and lots of leg means go forward NOW. Again, it's best to work with a trainer on these things as there is only so much random strangers on the internet can do, without knowing you or the horse.