Can a 'normal' headache, or headache caused by sinus pain turn into a migraine, if left untreated for too long?

Different animals: Sinus headaches are caused by increased pressure in the sinuses, usually by some sort of blockage. Migraines arise from different processes, though it hey are not fully understood. That said, untreated can sometimes lead to potentially serious consequences. So sinus infections that last for more than a few days or get worse should be evaluated by a physician.

Yes sometimes: There are some new insights into sinus headaches and their relationship to migraines. The cause of both is different but there is a definite association between the two. Often if the trigger for their sinus headches is determined and treated, patients state that their migraine frequency and severity decrease over time.