Yogurt was virtually unknown in the world outside of the Ottoman empire.
Even a hundred twenty years ago, there were European travel records about this mysterious yogurt which could be consumed in large quantities without adverse effect and which was so different from the already known sour milk.
During the collapse of the empire, a Jewish Ottoman resettled to Spain and took the yogurt with him. He would administer it to people suffering from gastrointestinal problems. Eventually this yogurt proved extremely popular and he founded a yogurt company named after his son that was born in Spain, Daniel who was known by the pet name of Danone.
The introduction to America happened similarly. Ottoman citizens (Armenians etc) that were fleeing the collapsing empire brought it to America and retained the Turkish name as Turkish was their lingua franca. Also comparable with pastrami from the Yiddish language which derives from Turkish “pastırma”, pressed meat.