Cradle cap and hair loss

My baby is 4 months and he’s had cradle cap for what seems like months. I bought come specialized cradle cap balm that’s basically a bunch of different oils. His hair has started falling out and it seem that the cradle cap has gotten worse- it looks like the cradle cap has been forming around follicles of hair. Today I really lathered it up and used the soft brush and large cups of his hair started coming out! I’m so upset about it.

Is the hair loss normal? I posted a photo of what his hair looked like about a month ago, he still had cradle cap but had so much more hair.