New Build Lawn Advice - 12 weeks in

Hiya, seeking more helpful advice from the kind folk on here.

Moved into a new build house in a housing estate, so as you can imagine the soil is probably not the best as builders wouldnt be going out of their way to deliver a perfect lawn.
The garden was seeded (by the builders) on/around 15th June (12 weeks) ago. I will add a few pics to show the stages.

So far, I have just cut the grass to 4cm on each cut and watered when the weather has been dry for a few days.
The house is built on old farmland.
I have noticed a good few creeping buttercup throughout the lawn & also what seems to be white clover & bitter dock also present.
There are still a couple of parts of the lawn when you get up close that are patchy and some of the grass seems to be yellow/brown (maybe from the sun).

1. I guess I am trying to understand what should I do next to improve the lawn?
2. Should I be adding any sort of feed to the lawn?
3. Should I be overseeding it?
4. Should I go around with the garden fork and aerate it?
5. Do I need to scarify it? I dont know what it is but have read about it a bit on here

All advice is greatly appreciated.

Pics below to show the change over time, 3rd pic of the lawn is the latest view.

 

You could put a autumn feed on it.

The patches seem to be getting smaller, you could put some seed in those, i think that you can do it in the autumn or it might be better in the spring.

It isn't established enough to scarify it yet.

We don't bother about weeds in ours.

The yellowing could be from the weather.

 

Looks like it got dry earlier but has grown back.
I dont fret about grass, just cut it when long and it always looks best in spring.

If you want a fine lawn that is different and lots of work.

 

Thank you, autumn feed and a bit of seeding sounds like a good next step.
Autumn feed before seeding?

Good to know about no need to scarify.

I have just read a few posts mentioning getting rid of the creeping buttercup where possible.

Thank you Pete. Yes, I would say that is probably what happened - got dry earlier.

I dont think I want a fine lawn yet per se and with the quality of the soil in the garden it would probably entail replacing the soil in the garden.
However, I would like to improve it as much as possible without a massive committment.
Autumn feed and a bit of seeding would probably help improve it overall for now.

 

I'm not sure which is best first and your welcome.