Cradle cap..

Ive been trying really hard to get rid of my sons cradle cap, it was super thick and caked on, and with a bath brush and doing Circular motions when washing his hair we now have a very flakey result, they are all lifted up I could pick them out but I don’t wanna do that.. how does that come out now? It looks awful. He has super sensitive skins so no oils or anything work for him, and he gets rubbed every other day sometimes every two due to eczema. Please be kind, I feel as though sometimes people can be really judgey and I’m just looking for positive feedback and advice from one momma to another!

UPDATE:

So after reading through peoples suggestions and comments I ended up trying some Vaseline and putting that on little babes head and let it sit for about 15 minutes then I put him in the tub and used some Aveeno baby shampoo and my Frida baby cradle cap brush set and used circular motions for a few minutes then got him out of the tub and use the little comb that came with it and most of his flaky scales came off we still have some last and now his hair is greasy from the Vaseline but I would say this is a win! I can deal with a little greasy hair for a few days!!