I'm sorry this happened to you, but if you drive long enough, wrecks aren't an "if" but a "when".
It's good that everyone came out of this one OK. I know of a navigator that possibly broke her arm at a CDE this weekend when the driver took a corner and the vehicle rolled.
Every horse has a "stressor reaction". My show horse is (was) a bolter, this one is a bucker. It doesn't necessarily mean that the harnessing is incorrect, she could have been stung by a bee, or decided that what she was doing was too mentally challenging, or she is indignant, etc. The trick is to catch the "pre-stressor reaction" before it gets full blown. This takes practice and horsemanship.
It's hard to tell from the photo, but the front hip strap seems a bit low. With the breeching more level, the slack would be taken up in the breeching and the potential for getting a leg over the shaft would be minimized. Yes, a kicking strap would help, too.
On this vehicle, I also wouldn't mind seeing the singletree lowered so the line of draft was more horizontal. See how the trace goes "uphill"? I doubt that contributed to this wreck, but it would help the vehicle pull easier. We have mounted the cross bar and the circle bar under the shafts upside-down and put the singletree on top. I do know of other vehicles that have mounted the whole assembly upside-down with the singletree on the bottom, too.
This is another challenge with "Easy Entry" carts. They are also Easy EXIT, and not always by your wishes. A bit of tufting on the vinyl seats, or cloth seats would have helped you stay with the cart. Some people also use "rubber" shelf liner under their butts to help hold them in place. NEVER bail from the cart!!! It's one thing to be thrown from it, but to jump off is not only dangerous for the driver, but that horse is dragging a weapon behind it. If you are in the cart, you at least might have SOME chance of slowing and stopping the vehicle! YOU put that vehicle to that horse, it is YOUR responsibility to see that OTHERS don't get hurt. If you aren't in the cart, you are still liable for whatever that horse runs into or wrecks, and you can't possibly stop it from the ground. STAY WITH THE CART if at all possible. (Written for the masses reading. I know the OP was thrown.)
Also, unless you intend on showing in mini breed shows, I would dump the overcheck. The mare is built very level and needs the ability to lower her head to move the cart. The overcheck entirely prevents this and actually pulls the horse's nose up. A loose overcheck is a danger, as those loose straps can get under the shaft if the horse puts it's head down, and then tighten enough to make the horse rear. A sidecheck can be more beneficial, but in actuality you don't need it. Obviously, it didn't help keep her head up to prevent bucking. Most people use it only because it came with the harness, but really don't understand it's purpose. In the big scheme of things, I doubt that some of the Amish that build harnesses know what the purpose for a check is!
That is just the "way it is done". That is not a good enough reason for me to use a piece of equipment.
Keep Driving!
Myrna