That window "frame" is basically a thin piece of material that is used to trim out the window. I don't know how thick it is, but behind it is likely lumber framing. The screws that you use should go through that trim material and in to the lumber framing for the strongest connection.

You can use normal wood screws or perhaps even better, interior construction screws. Just stay away from the black drywall screws. They're brittle. Calculate a 1/2" for the trim material and at least an inch of embedment in to the framing lumber behind it. So that's a 1 1/2" or 2" screw. A pan head screw might look best.

The issue you have with splitting here is that you did not pre-drill the hole. Trim material is often a bit weak/brittle, especially toward the end of the board where your screw is. The drill bit size that you use should be about the same size as the shaft of the screw you want to use. NOT the full size of the screw and it's threads. If I’m doubt, make the drill bit size a bit smaller than a bit bigger. Your pre-drilling would only need to go through the trim material. You do not need to pre-drill the stud behind it as well. Though you are welcome to as it won’t harm anything.