You shouldn't think the bitless or bit is a hand brake.
You NEVER ever should pull hard on both reins regardless if you're using a bit or not. That is why your mare pushes through the pressure. You've trained her to do that, by pulling on her head.
You're riding with your hands. That's totally wrong. That's how jockeys teach race horses to push through the bit pressure. They pull hard on the reins, hold that pull so the horse is running hard, but not as hard as it can. Then when they release some of that tension, the horse runs faster in the final stretch of the race.
That's what you're doing. You've taught your horse to ignore the pressure, because it means nothing. You're just reactive not being an active rider. And that is the real problem, not what you're using.
I strongly suggest taking riding lessons from a trainer so that you learn how to use your seat and legs vs just riding with your hands.
What you can do right now, though is practice this simple exercise to retrain yourself and retrain your horse to respond to pressure:
1. go forward at a walk (with the rein on the mane. one hand on the rein only)
2. you'll ask the horse to stop by:
-- exhaling
---rolling your seat forward....squeezing your butt cheeks (this is called using your seat) hold the squeeze --this tells the horse you have stopped riding
--pick up the rein in the middle (use a loop rein, not split reins) with one hand only (raise it high) and run your other hand down the (left) rein, follow the outside seam of your pant leg. Until the horse bends the head around toward your boot toe.
--press your (left) leg against the horse's side to ask the horse to move the hip over (the left hind foot will cross in front of the right hind foot)
If you pick up enough rein, the horse should be bent around your (left) leg and should stop forward movement, only moving the hip around....then stopping completely.
When she stops completely and isn't pulling against the rein, let go of the rein and just sit there for the count of 5.
3. Repeat with the other rein. (right).
4. Once you can do all the steps easily at the walk, repeat them at the trot.
5. Then repeat them at the lope, but spiral down, don't just bend the horse around in a hurry.
This is the one rein stop.
If you exhale and sit down (squeeze your butt cheeks) and hold that before you pick up the rein. The horse will learn to "rate" your seat....meaning, she'll know you have stopped riding, so she'll stop.
This is using your seat properly.
You use your seat to stop the horse, not your hands.
Here's a youtube video with Clinton Anderson (awesome trainer) talking about horses that have too much Go:
http://www.youtube.com/watch?v=NVLAVNvSj8w&feature=related
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By the way, I ride with a Dr Cook's Bitless....horses that have too much go, horses that have run off with their owners. So, I know that you can retrain a horse to stop running off with you or pushing through the bitless (or bit) pressure.
It's not what you use, but how you use it.
