Do horses get epilepsy?

Yup. It's pretty rare, but equine epilepsy definitely exists - and a 1000 pound animal having seizures is dangerous to everyone.

Epilepsy can be caused by brain tumors or infections and some parasites. It can be managed in horses with modern medicine and some foals that have epilepsy grow out of it.

The Arabian breed carries a genetic fault that causes neonatal epilepsy - fortunately, these foals almost always grow out of it by the time they're yearlings. These foals have cluster seizures and can sometimes hurt themselves pretty badly - but they grow into normal adults assuming they can be kept from, you know, banging themselves on the fence too hard.

There are anti-seizure medications for horses, but (as in humans) no cure - and horses that suffer seizures regularly should, obviously, not be ridden or driven.