The "Turkish" hip twist is a horizontal movement, with the hips swiveling front and back. The navel moves with the hips. There is no up and down movement in the hips. In my dance experience, there are two very different moods that accompany this particular movement in the hips. In a more "Turkish" style or gypsy style dance (as always be aware that these labels are always a bit arbitrary) the movement is smaller and faster. In a more folkloric or Egyptian style version. the feet may be wider with the movement resting more on one side.
The video available below from Delilah of Seattle's Visionary Dance Productions is one of the first videos I used when I was learning to dance. Her videos are classics for learning! This particular series is one of the first of its kind from Delilah's Belly Dance Workshop Series.
The series: Shimmies: A Source Book.