New lawn not looking well - need help to rescue

[momo2007]

Hi all,
Need some advice how we can hopefully save our new lawn that was laid by a company on a bed of top soil and fertilizer in end of September. As advised we watered it every day apart from when it rained for 4 weeks. Lawn looked great, grass grew a a lot. Then we mowed lawn perhaps too much.

Now, as pics show we have a lot of flat brown patches where turn edges are visible, which I think the grass is dead. The turf is still liftable and their is hardly any white roots to see.
Most of the brown patches were probably caused by me walking around (after 4 weeks of using wood planks to water lawn)which I don't know how you would avoid and also probably by all the leaves that fell and were not removed early enough.

Any suggestions on how to recover the lawn? Not sure why it still has not rooted? Should I start spreading lawn fertilizer now in coming January? Or seed all the brown flat patches?

Advice appreciated.
Mo

https://ibb.co/S3jYKCK

https://ibb.co/1QkM46Z

https://ibb.co/Fgkd1cY

https://ibb.co/sgkk25s

https://ibb.co/ZgS42Qf

https://ibb.co/FhGGNVN

https://ibb.co/X4fzx1k

 

[momo2007]

Sorry, that's why I got confused when asked for pics when I thought I had posted pics!

 

[momo2007]

Yes, around the tree I left it for a while and that's where it has a lot of bare patches. I started raking it when it was still wet and damp which probably didn't help. I left it to grow as advised after leaving it for 4 weeks, then cut it twice