It looks awfully similar to the rash one of the rescue dogs had, and was attributed to a flea allergy from before we took her in. It got better eventually, in the meantime she was given benadryl and apple cider vinegar (unpasteurized) was added to her food. So even if your house is flea-free, she might just still be recovering from it.
Also, it’s totally fine to keep a dog on puppy food for the first year or so.