Equiverse

Summary
From website:

Equiverse is a competitive horse sim game in which you can create, breed and show your very own horses! Here, we encompass the realistic aspects of owning horses - from caring for them, breeding them, showing them, and much more.

You can do anything from building up your empire of Show Jumping Arabians, to breeding your Thoroughbred Race Horses, to collecting an assortment of colored Appaloosas. We boast 20 different breeds of horse, 6 different disciplines to train and show in, and over 2 million different genetic combinations - always with the possibility of new genes being discovered!