Context: A young adult tells his mother that he’s depressed and that life is meaningless and that he wants to die (suicide is not directly mentioned but implied). Then the mother proceeds to express “regretting giving birth to you” directly to that young adult.
(That young adult is me)
That is incorrect
https://www.merriam-webster.com/dictionary/normal
First definition from your link:
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”.