"Cast on" feels like the wrong term for what's happening with the Turkish cast on because you're really just wrapping the yarn around the needle. Even as I say this it will feel like you're doing something wrong because you're doing so little, but that's really all you do.
Take the yarn from under the needles around the back and over top of the needles and back to the front of the needles and so on, looping around both needles as you go each time.
Repeat until you've made enough loops to account for half your needed stitches, minus one. In this example we were going for 20 stitches, which means I needed nine loops.
Wrap the yarn around the top needle one more time and bring the yarn between the needles. Now there are 10 "loops" on each needle (counting the slip knot on the bottom needle).