Start by learning the thumb cut.
What helped me was to use both hands. Use the other hand (the one that's not going to do the move) to set up the deck in a position you can do the move. Worry only about completing the cut. It will take some time and practice, but eventually you won't rely on the other hand and it'll become smoother.
Another thing that can help you is using a bridge deck, which has smaller cards; or binding packets with elastic bands.