Shaving babies head?

My sons hair is falling out and it looks terrible! He was born with a MOP of copper hair, so his hair loss is super noticeable & people comment on it. He’s only three months so I know it’s normal, but people who don’t have children & random people in public always say something about it 🙄

Has anyone ever shaved their little babies hair? Like completely bald or buzzed? How old were they? Did it grow back?

I’m driven nuts by it being patchy 😭