And the bigger question? What is the soil like underneath the current lawn? That'll suggest remedies that may work!
I'm guessing from the photos - but could easily be wrong. That the turf roots form a mat maybe an inch thick; don't go much below that? Probably a clay soil, with a "thin mat" of turf? Which ..... well, it basically slides around on the soil surface because lawn grass roots aren't strong enough to bore more deeply through the soil.
Peel back a wee bit of turf, and take a handful of the soil. Squeeze it into a ball with your hand. Then - gently break the ball.
- If it breaks and crumbs really easily, then I'm wrong, and it ain't a clay soil.
- If it sticks together, holds its shape, and feels kinda sticky - then it is a clay soil.
Do that wee test - tell us what your soil ball (!) feels like.