It REALLY depends on the kid.
People who ENJOY horror movies (note the key highlighted phrase) aren't afraid or disgusted by them. They might get a little tingle of fear, but they don't have bad dreams, panic attacks, vomit, etc.
All of which my son got from Disney movies. He was scared witless by them (the evil guys in Disney don't have ANY redeeming features... we're talking baby killers -snow white, lion king-, sociopaths -little mermaid-., your parents being murdered (too many to list) all set to jazzy catchy music. Um. Yeah. Okay. It took my son wigging for me to SEE that, but I can definitely see it.
PG13 and R movies (not all of them, of course) on the other hand didn't bother him at all. The complexity was relaxing to him. And that the heros and heroines were both flawed AND not universally stupid/only surviving via luck, meant that we did a lot of THOSE movies.
So, to me, there's no "age" for any movie. It's when and what scares someone. Really, truly, scares them.
We don't do horror films (at 9), just because *I* don't like them. My son would probably be fine with them.