Leave it alone for now, just keep mowing little and often, and collect up the clippings if you can. The new growth looks healthy enough. Once it's well established (maybe next year, but other more knowledgeable lawn experts on here will be able to tell you) you can rake/scarify to remove the thatch, which is buildup of dead grass blades in the lawn, and is normal.
I wouldn’t fertilize a new lawn ... as had been said I think it’s looking quite good. All lawns have some dry grass at the base ... when it builds up you can take it out but your lawn isn’t established enough for that treatment yet.
Hopefully @glasgowdan will see this and offer a lawn expert’s advice.
“I am not lost, for I know where I am. But however, where I am may be lost.” Winnie the Pooh
You water bill is going to be HUGE, (ideally dont lay turf in the summer, too late I know, )dont apply feed, it will probably burn it, unless you have heavy rain, you will need to water it, it wont have rooted in after a month.Hubby removed grandkids playhouse, laid turf, we had quite a bit of rain, its pretty shady, under a eucaliptus, I missed watering one piece, right by the edge, under shrubs and YES it has died, am on the naughty step!!
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Hopefully @glasgowdan will see this and offer a lawn expert’s advice.
I think it is getting worse now: What should I do? Please help. I was told @glasgowdan is a pro?
Which way does the lawn face? How much sun/shade does it get?