The short answer
Kuşadası has no airport of its own. The closest is Izmir Adnan Menderes (ADB), 79.5 km by road and about an hour and a quarter of driving. Milas-Bodrum (BJV) is the next option at 111.7 km and roughly two hours. Dalaman (DLM) is 249.3 km, about three hours and forty minutes, and Denizli Çardak (DNZ) is 249.5 km, about three hours. For almost everyone the answer is Izmir.
