People mostly didn't ride horses, they had the horse pull wagons but they rode behind. You rode a horse where speed was needed, but that meant you have a series of places to trade out your now-tired horse.

Even horse above is wrong - you probably had oxen to pull the wagons not a horse. Oxen ate a lot less and where a lot easier to work with in general so they would have been preferred. Mules, donkeys, or even goats may have been used as well. There are pros and cons to all choices, but in general the horse was the most expensive and used only where it matters.

The horse was used in the American west above the others though. The prairie soil needed a plow that was pulled faster than the others, and so a horse was needed to break the ground. Cowboys road a horse because in the case of a stampede a horse was tall enough not get you killed (assuming you stayed on it), and fast enough to divert the herd. The above is specific to the situation in the American west and doesn't apply elsewhere. (note the that cowboy stampede situation is similar enough to knights in battle that both would use a horse despite the disadvantages)

The reason we mostly think of the horse is once the plow caught on in the America west it was enough better that much of the rest of the world started adopting it. This needed the industrial revolution to be under way though, and of course the tractor and automobile were not far behind. Before the industrial revolution were thousands of years where the horse was a rich person's toy, most didn't have them and if they had animal labor in reach they would want something else over the horse.