Zelda Breath of the Wild – How to Get More Ancient Arrows

Having more endurance or stamina should help you, as you get thrown off when you run out of stamina. The easiest way I've found was eating energizing foods while taming, since it's possible to stop whatever you're doing to gobble up something. You can cook a few Stamella Shrooms on their own or make an energizing elixir, anything that will restore more than a quarter bar, since you want to have time to actually soothe the horse. You can then fill your stamina bar when it gets low, which will allow you to keep soothing the horse until it is tamed. Make sure you have several of those on hand; some horses can take 3 full stamina bars to calm down.

If you have access to enduring foods or elixirs, they should work well too, as your stamina won't go down as fast, giving you more time to work with. However, you may still want to have stamina restoration consumables on hand.

The only way to fix it would be to load a previous save and fight the boss once again. I know it's a struggle, but right now its the only way.