Turkish was written with an Arabic-based alphabet until 1920s. The move to Latin alphabet for Turkic languages is new, and not yet universal. For example, Azerbaijani, the closest Turkic language to Turkish (so much that Turkish and Azerbaijani are mutually comprehensible) is written in Latin alphabet in the Republic of Azerbaijan but in Arabic alphabet in Iran.
Yep! Orthography is hugely political for Central Asian languages, so I am really disappointed that this article would make such a glaring mistake, among others. It does not seem very well researched.