Cradle Cap! Help and advice

My 11 week old has quite bad cradle cap. Mostly I’ve left it completely alone and been applying coconut oil but the scales still remain at the front and back of his head. Today I washed his hair in the bath and used a soft bristled baby brush and rubber flat brush in a circular motion to try and wash away and loosen some of the scales. When it’s wet they don’t come off very easily so dried his hair and brushed a little more. It seems to have caused some irritation and redness so will be leaving it alone for a a few days before attempting to remove the cradle cap further. I’ve used a generous amount of coconut oil again to soak overnight and applied some of my breastmilk on it (which is a miracle cure for most things) I’ve ordered special cradle cap shampoo from Amazon and will see how that goes before visiting GP for advice. Any advice appreciated 🙁