Please help me with my lawn !!!!!

hi there, right heres the story bought my first home from my great aunti who always had a lush lawn she sadly passed and just under a year later we started work on the house at the end of january.
I was in there every day/night and didnt have time for the garden.
it got near the end of the restoration and i had spent liturally a full week in the house from 8am-11pm and me sparky let is down for the second fix and i thought av been in there for that long am not doing no more painting and decided to tidy the garden removing all rubbish and tackling the lawn which was roughly 6-8 inches and full of dandilions, yellow heads and the seedy types.
the first cut was in may on the royal wedding bank holiday and it was a nightmare but when it was all done and dusted it looked great and thats why i am now becoming obsessed with the garden :yess:.
so to tackle the weeds i bought some weed n feed and applied roughly at the end of may it seemed to do the trick clearing weeds and greening the grass but it did scorch a few bits since then i have cut every week keeping the weeds down but we had allot of rain and sunshine when the hopping was here the grass grew like mad along with the weeds so i have tried removing weeds and roots but there is to many i have aftercutted twice but i am wondering if i need to weed n feed again and miracle grow the scorched bits. i have bought the after cut easy dispenser as the coverage is the same as the weed n feed so hopefully wont scorch like i did the last time. my mate keeps telling me GREEN THUMB !!!! but i really want to learn to tackle this problem my self as i am really enjoying gardening.
at 24 the garden and presentation of my home is more important than the bar lol.
sorry for the essay but just trying to install as much info as possible and any help is appreciated and thank you in advance.

 

I've tried weedkiller on lawn only once .... followed instructions but found that some of the lawn was killed aswell

so now, I just use a knife and remove weeds with as much root as I can ....