The “-mare” in nightmare is a demon, not a horse
The scariest part of “nightmare” isn't the night. It's the mare, an old word for a creature that sat on your chest while you slept.
The mare in nightmare has nothing to do with horses. It's a much older, much creepier word: a demon.
The etymology of “nightmare”
In Old English, a mare (or mære) was an evil spirit — an incubus that crept in while you slept and crouched on your chest, crushing the breath out of you. A nightmare, originally, wasn't a bad dream at all. It was the thing: the monster doing the crushing.
And it wasn't just a story. That suffocating, can't-move, something's-in-the-room feeling has a name today, sleep paralysis, and it's common enough that nearly every culture invented a creature to explain it. The English one was the mare. Related demons haunt witch lore and old Germanic tales, all doing the same thing: sitting on the sleeper.
Only around the 1500s did the word slide from the demon to the feeling it caused, and from there to any bad dream. So the next time a nightmare wakes you at 3am, blame the grammar: you're not having a bad dream, you're being visited.