Do stallions fight over mares?

Horses are harem breeders, meaning that only the dominant males get to keep a "harem" of females and breed. So, yes, stallions do sometimes challenge each other to get to be the dominant/breeding stallion (the rest of the males live in bachelor bands). However, not all wild horse fights are over females - sometimes they can be over water rights, a particularly good patch of grazing, etc. And both mares and stallions in bachelor bands will have dominance disputes.