If it's itchy, it could be eczema at this point.
The child might need to see a dermatologist at some point. My kids all have had it - it is worse in the winter months. One of my children has it very badly.
We've tried CerVae cream as mynewnickname suggests (recommended by our dermatologist) too - that is nice and thick and will stay on.
So cortisone gets rid of the red scaly itchiness, and you'd get a prescription (or at least we do). You might be able to get over the counter stuff, but likely not high enough a % to be effective. The thing is, you only use for 2 weeks twice a day to get it under control.
So we use a tub of this thick white paste which is just given out at pharmacy (no specific brand, they make it up) called 'hydrous emollient'. It's mostly water based, but thick white paste. You can ask them to make some up for you. You coat them in it and it's for kids with chapped, irritated or eczema skin. It creates a barrier and is soothing. My kids love it. It rubs in.
We also Aveeno lotions too for very dry skin.
We use lotion that is creamy and aloe vera based at home (not the clear gel) and nothing extra sanitizing. Or Dove unscented bars.
Sometimes eczema (if that's what it proves to be - if it persists or shows up anywhere else, like arms, knees, etc.) can be allergy related - as in the case of my son. Do you have a cat/dog? My son had it bad at that age because we had multiple animals.
Good luck :)