Is it wrong to have a girlfriend at 20?

Its totally upto you. If you think you need to have a girlfriend you might search for the right one but staying single is cool too,and way better than being in an unnecessary fake relationship which is temporary. Do what you like,even if the society demands you to have a girlfriend at 20 years of age,so be it.

Is it weird to start dating 21?

Starting dating in your late twenties isnt as uncommon as you think. Theres nothing to be embarrassed about and, no matter what, dont settle for the first person that comes along just because youve been waiting so long. You deserve better— and youll find it, whatever age you start.