First definition from your link:

conforming to a type, standard, or regular pattern characterized by that which is considered usual, typical, or routine

I’m guessing the other person meant “the way it should be” not in a normative way but as in “the way you’d expect it to be since that’s the usual way”. That’s in line with the definition. Bottom line is both have several definitions but “normal” definitely has a very different connotation from “common”.