Sounds to me like you’re trying to loosely wrap the string until it catches and holds. That promotes slippage. Use the loop around your finger method you’ve been using, or the pinching thing andy569 described, but do more wraps than what you’re used to doing with a smaller gap. Keep the wraps tight, and don’t pull your finger out, or let go of the pinched string, until you can keep wrapping tightly without it slipping.
At some point, you’ll definitely want to learn other methods of starting like andy569 suggested, as winding while the string is attached to your throw hand, creates string tension (twists the string).
https://www.youtube.com/results?search_query=unresponsive+string+tension