Should I see a specialist for oral thrush, or can my regular doctor take care of it?

Yes. It is always good to have a diagnoses , to rule out any thing serious and i agree with other physician to see u pcp for evaluation and treatment.
Regular doc. Oral thrush is common in newborns, but after that time it should be not only treated, but a cause should be evaluated. Certain drugs, such as antibiotics, steroids, and chemotherapy, can predispose to thrush. Serious medical conditions (immune problems, diabetes, etc) can be identified with a symptom as innocent as thrush too, so please have your doctor evaluate!