Are you training western or English? A western saddle gives you a little more to hang onto if the horse decides he doesn't like you there on top of him.
If it is western, there is a proper and safe way to mount that is becoming less and less known nowadays. It was developed and practiced by old-time cowboys who, often as not, rode half-broke broncs on a daily basis.
When you mount, take the reins in your left hand. Snug them up so you have some control. Some like to pull the horse's head toward you a little - shorten the left rein - so the horse doesn't have the freedom to pull his head down and get his hindquarters engaged to buck.
Now take up a big, solid handful of mane with your reins hand (left) just ahead of the withers a little...maybe a foot ahead of the saddle.
Stand by the horse's left front leg facing the rear (to keep from getting a cowkick in the pants) and put your left foot in the stirrup. You have to twist it around with your right hand.
Face the horse and grab the HORN, not the cantle, with your right hand and pull yourself straight up in the stirrup (this is after all your testing and you are really going to mount). Use the hand with the mane hold to pull a lot of the weight, so as not to pull the saddle over.
Now, swing your leg over and sit in the seat.
Mounting in this way makes it so that if the horse decides to bolt or buck while you are mounting, you can still continue into the saddle or you can stand in the stirrup, holding the horn with your right hand, let go of the mane and pull the reins with your left. It gives you the option of continuing into the saddle, staying in one stirrup, or stepping back to the ground.
If you mount with a hand on the cantle, you cannot continue into the saddle without letting go and you are left hanging onto the side of the saddle until it slips off to the side and dumps you on the ground, or the horse rubs you off on a post, possibly with a foot still in a stirrup. The only safe thing you can do in that case is dismount. Horse wins.
For a great look at what it looks like, watch any John Wayne or Tom Selleck western.