I'm no turtle expert, but I find it doubtful that he would be shedding for over a month (you mentioned in your photos that you first noticed the white spot a month back). What is your basking set up like? Water parameters? I would definitely want to thoroughly clean his set up. It might be time to seriously consider that upgrade you've been planning for!
Its hard to tell from the photos, but the white lesions (particularly the one) looks like it is definitely concave. Is the area foul smelling? What is his behaviour like? Is there a chance you've been over-feeding him? Have you noticed any other questionable areas (how is his underbelly (plastron) looking?)?
I'm not sure if you've already initiated a treatment regime. I found a very good article
on the home treatment of mild rot. I'll repeat a bit of it below, while trying not to sound like the efforts of a dying record player!
Have you tried giving him a good clean using a soft (unused!) toothbrush with some ever so gentle diluted soap? Afterwards you should be able to use the edge of a clean (hit it up with some rubbing alcohol or boiling water if you don't have any on hand... NOT hydrogen peroxide!) credit card (or empty gift card etc) to gently scrape at the white areas. Don't dig at it or anything... just remove anything that comes off easily.
You then need to disinfect with either a mild saline solution (aquarium salt, kosher salt), I know others use potassium permanganate, or the Nolvasan antiseptic discussed in the aforementioned online article. You then should keep him dry for at least a couple of hours and repeat disinfections each day for a week.
If you don't see any improvement, his symptoms worsen or you are really uncomfortable/unsure of treating him you should consult a veterinarian. Ideally a reptile or exotic vet, of course, but realistically all vets should have some background in dealing with similar topical conditions.
Best of luck to you!
PS: Welcome to the community! I'm also new to the site (but not the hobby).