I'm not great on Turkish history, but I have this idea running through my head and I guess I'd like to throw it on the slush pile.
Atatürk stops drinking. He lives for another fifteen or so years, seeing Turkey through WWII and beyond. He also keeps Turkey neutral, but he sways towards the Allies after seeing how Britain especially could be a model for a democratic Turkey. Now, after WWII IOTL İnönü democratised quickly out of fear that it'd look bad if Turkey was a dictatorship and not a democracy after the war. Atatürk, however, doesn't do this, especially since the Allies aren't exactly committed to toppling the likes of Spain and Portugal after this, so he seeks guidance from the Allies on properly democratising Turkey without risking reactionary backlash.
Essentially, the CHP in Atatürk's twilight years becomes a pragmatic conservative party influenced by the likes of Adenauer's CDU and Churchill's Tories. They form the basis of the Turkish welfare state (more along British Labour lines), but they aren't dirigiste, especially since Celâl Bayar is influencing economic policy. As such, the left wing of the CHP splits off, forming a social democratic party in opposition to the postwar CHP's nascent conservatism. Thus, Turkish democracy is born naturally.
Atatürk dies around 1953, and if everything goes to plan, there aren't any internal military coups ITTL. That's the ideal, anyway. God knows if that's how it'd work in practice.