Since this horse is not yours, you are just taking lessons on him from your trainer, there probably is not a whole lot you can do training wise.
However rememeber when fencing and stopping, do not anicipate yourself, your horse will read it. When fencing, look beyond the fence like it is not there and let the fence stop him, not your anticipation of the fence.
If you are getting tipped up when stopping, either your horse is dumping his front end or you are stiff(anticipating getting tipped forward can cause this) and not sitting correctly. Perhaps you could get a video of you stopping and get a look at your position.
I DON'T LEAD 'EM AND FEED 'EM, I RIDE 'EM AND SLIDE 'EM.