HAVE YOU BEEN INJURED? I HAVE BURNING IN MY RIGHT FOOT. HAD A NERVE CONDUCTOR STUDY. WAS DIAGNOSED WITH COMPLEX REGIONAL PAIN SYNDROME, A RESULT FROM FOOT SURGERY. IF YOU HAD A INJURY TO YOUR LEG IT WOULD BE WISE TO SEE A NEUROLOGIST. SOMETHING AS THIS SHOULD BE DIAGNOSED ASAP. I HOPE THIS HELPS IN FINDING A ANSWER
That sounds like something that can drive you mad! I have on occasion that same kind of thing on my back. For me, it is related to dry skin I suppose although it doesn't look like it. Once I start itching, it's everywhere on my back. I got some eczema balm made by Aveeno (got on Amazon or I believe CVS carries it and maybe Walgreens and walmart) and that makes a world of difference. I put it on after I get out of the shower. It's thick. And it really makes me not have the itching situation. Not sure if this is the same thing but something to look into. https://www.aveeno.com/products/eczema-therapy-itch-relief-balm There are some severe leg itching things that are related to exercise. They can either have to do with exertion and capillaries in the legs opening up or allergens that are exercise induced but this doesn't sound like the pattern for you.
I'd try to use lukewarm water rather than hot, try the eczema balm and if that doesn't solve it, see your doctor. Some blood issues can cause this type of severe leg itching.