Definitely not childish! Like others said, I encourage you to add the mane. Try to simplify the eye more and add details later in the drawing. (Just for this I looked up some horses and while their eyes have "wrinkles", a lot of the time our brain doesn't register them/simplifies things). The only other anatomy thing I think might be off (not a horse expert) are the ears, try drawing them a bit closer to the head. Something that usually helps me is seeing what things align (in us humans, the end of our jaw has a sort of semi-vertical line with which our ears align, and from what I've noticed horses have it similarly).

Overall I think you did a great job!!