They used to use a modified version of the Arabic alphabet until 1928, when the alphabet was changed to a modified Latin alphabet. Mustafa Kemal (later Atatürk), the leader of the Republic of Turkey, felt that Turkey needed to look toward Europe as a model for a secular republic with a religious population (one of the criticisms about why the Ottoman Empire failed was that its leader was both political and religious, and that this was best avoided).

It can also be argued that the modified Latin alphabet fit the sounds of Turkish better than the modified Arabo-Persian system used earlier (the original Turkic writing system looked more like the one used to write Uyghur, although it is questionable whether the groups that settled what is now Turkey ever used it).

The language reforms also were meant to remove foreign loan words from the language and replace them with “purely Turkish” words (Turkish is not linguistically related to Arabic or Persian, but had incorporated a lot of words from both).

The language reform worked because most of the population was completely illiterate (in the high 90s, percentwise) and so it didn’t particularly cause a lot of upset because the vast majority couldn’t read the existing alphabet anyway. In tandem, a national educational initiative was pushed out, bringing teachers into small villages to teach people how to read and write using the new system.

Turkey’s literacy rate a century later is quite high, but as a result of the reforms, a lot of Turks today can’t read anything written before 1928.