let the player move first
A camera deadzone is a box in the middle of the screen where the player can move while the camera stays still. When the player pushes past an edge, the camera moves just enough to keep them on that edge, so small steps and turnarounds leave the view steady. Here the camera holds still each time the player turns back and crosses the box.
Give the follow camera a deadzone: keep the camera still while the player is inside a centered box about a quarter of the screen wide, and when they cross an edge, move the camera by exactly how far they went past it.