The Arabic writing system is an abjad, not an alphabet. The two kinds of system are closely related (the Phoenician script was the origin of both the Greek alphabet and the various Semitic abjads and was itself an abjad) but are not the same thing. Abjads are well-suited to Semitic languages where the vowels are less important for morphological reasons, but in Indo-European languages (like Greek) and Turkic languages (like Turkish) vowels are important in writing for comprehension. It's no surprise that switching to an alphabet aided literacy in Turkey.

Abjads are no more or less complicated than alphabets though. They're just a bad fit for Turkish.