Try not to use the edge of your fingers if possible. When you roll your finger to make a tight turn, the game may rule this as a lifted finger and stop your progress on the puzzle.
You can't begin a puzzle until all movable blocks are touched simultaneously. Take a few seconds to position your hands into a position that feels comfortable before starting off. If a position feels painful before you start moving your fingers, it will only get worse as the puzzle continues.
Don't hesitate to play this game with a friend - many of the four-block puzzles are much easier to play with an extra set of hands on them.