One factor to be aware of is the strength of the "yes" campaign abroad. To quote an article from a month ago:
Turkey’s president, Recep Tayyip Erdoğan is campaigning to greatly expand his presidential powers in an impending constitutional referendum. To win, he needs votes from the huge Turkish diaspora. His government is going after them with gusto – and in the course of just a few days, those efforts threw up a fully-fledged diplomatic fracas.
This "diplomatic fracas" may be another factor in itself. Both in the Netherlands (see the article quoted above) and in Germany, local officials intervened in campaign activities, citing security concerns. Decrying these measures as "Nazi practices", Erdoğan played this interference up in his favor.
Finally I would point out that the split in support for the referendum closely reflects the split in support for Erdoğan himself. He is a charismatic leader, and the people who voted yes were essentially voting their confidence in him.