Supernatural is somewhere between horror and fantasy, basically.
If it takes place in another world that doesn't have recognizable events and figures from the history of our world, and where magic or magical creatures are significantly more common than our own, it's probably fantasy. This is also called "second world" fantasy. If main character(s) go from our world to another world, it's portal fantasy / isekai. If things are pretty bad, gritty, pessimistic or violent in the other world, it's dark fantasy.
If it takes place in a world that basically resembles our own world, but with magical or mythological elements added, it's more likely supernatuaral, but could be fantasy.
- if the added magical/mythological elements are drawn primarily from scary stories, folktales, and the myths that precede those things, with elements like ghosts, vampires, werewolves, witches/warlocks, undead, etc "added to our word," it's probably supernatural
- if the added magical/mythological elements are drawn primarily from fairy tales that aren't as scary, or from over-influential fantasy sources like J.R.R. Tolkien books (which feature dwarves, elves, dragons, wizards, magic swords, orcs, etc all in antoher world) or Dungeons & Dragons (which is basically Tolkien + other mythological stuff thrown in) and these elements are more prominent than the supernatural stuff listed above, then it's fantasy
- if most people don't know about or don't really interact the added magical/mythological elements, so their lives are pretty much like people's lives in our world, then it's almost certainly supernatural
- if most people don't know about these things, and the main character(s) also don't, and they find out about it and horrible, horrible things happen to them, then it's closer to horror than supernatural
- if the main character(s) ARE magical/mythological creatures that most people don't know about or interact with, then it's much more likely to be supernatural than horror
- if average people can use magic, it's probably fantasy or sci-fi-fantasy. If there are schools where you go to learn to use magic and become a wizard or whatever, it's probably fantasy. If magic is unpredictable and dangerous and can't really be taught in a structured way, it's more likely supernatural. If using magic is rather likely to end up with the magic-using character getting turned inside out and screaming forever in agony and going insane, it's horror.
There are a bunch of things that could go in either category. Mermaids? If they're everyday creatures average people interact with, it's probably fantasy. If nobody knows about the secret mermaid kingdom under the ocean, it's probably supernatural. Faeries? If they're happy go-lucky friendly creatures who grant wishes but aren't main characters, it's probably fantasy. If they're a secret type of creature who are more devious and political, who occasionally interact with normal "our world" types of people, and the main character gets into some kind of relationship with one (or is part faerie) then it's probably supernatural. Undead liches and skeletons? Probably fantasy. Undead "I got hit by a car but it turns out I'm still here, just dead now" zombie? Probably supernatural, or horror if things get really awful.
