Oozing and itchy cradle cap

My 3 month old has a bad case of cradle cap that also seems to make her scalp extremely itchy.

I took her to the pedi who said it's normal and there's nothing I can really do about it but said I can soak her head in baby oil, massage with a soft-bristled brush to get the flakes out and then shampoo regularly. I did this for a few days and each time, she would ooze this thick liquid from her scalp that left her hair matted and stained everything yellow. More flakes and crust would "grow" back afterwards.

I took her back and again, her pedi was unconcerned and told me to leave it alone but my baby constantly claws at her head and will pull her hair out. She can't sleep due to the itchiness and even with her arms swaddled, she constantly turns her head left and right to try and relieve the itch.

It really hurts me to see her in so much discomfort and I feel like I need to try something to help alleviate or remedy this.