Is it normal to care this much to what others think about me?

I thought it was normal to care. Although, I think a lot of people try to pretend they don't. I only care about what the person has said if I respect them and I feel like what they said was accurate. I don't pay much attention to people who say things with the intent of being mean, e.g. trolls who say insults which aren't even true, why should I care? They're just doing it to get a reaction, why give them it? But if someone who knows me well and who I like gives me criticism, then I'll notice that.

I don't think you're strange for being the way you are. Sometimes it's important to "give a fuck" about what people say about you. Especially if it's constructive criticism, in which case not caring about that might stop you from improving. Sometimes acting like you don't care at all can seem a bit defensive or even a little bit arrogant at times.