This question comes up every year about this time when people are planning their trips out west. It is a valid concern and usaly less experianced riders ask this question (and its a good one). There is little out there as far as info or things to help the rider carry their long bow or recurve.
I know I may have people come back and tell me diffrent but I dont care its my oppinion. First there is not a real good answer for carrying such a long item. But I will say this by holding your bow in hour hand or straping it to you is by far the worst way. It is a acciedent waiting to happen it just may cost you your life PERIOD! I dont care how good of a rider you are or how big your ego is it is just bad. I will not argue I am simply stating what I have learned professionaly.
Just with getting on a horse comes risk. You start adding objests to get in the way just ups it. I had searched high and low and could not find any one building scabbards for long bows or recurves. So I started designing my own. The one pictured is the prototype and it has served well. I still use it. My new ones have zippers instead of snaps.
These bow are hard to ballance do to there light weight but I come up with what I feel is the best. I first worried about the tip of the bow hanging lower than my foot being a problem. I have not seen any thing go wrong if you hit a stump it just pushes out of the way. You can raise it but you run the risk of hang up and it makes mounting and dismounting more dificult. The quiver is covered so even while walking your horse through the timber I have had no hang ups.
When you are in rough country you need both your hands. Wrecks happen and it is split second. If you think your going to save your bow and controll a horse tell the outfitter you prefer to walk in. It just aint going to happen.
Arrow flynn what you just explained can happen to a seasoned horse or a green horse. Horses for thousands of years are accustomed to fleeing when danger presents itself. We will never breed that out of them. Some horse are better than others and handle situations different just as humans do under stressfull situations. Your horse has a mind of his own and you just might not be on the same page! Never attempt to shot from horseback and always get things situated with your horse before jeking your bow. I mean your off he is tied up or some one is holding him.
Again I will not argue this subject I feel I left some sound advice. You do what you see fit cause your going to any way. Good Luck!