Agricultural strength vinegar or environmentally friendly steamer. The hand held kind you use to sanitizer your house.
The vinegar will kill most anything. So if you want to spot treat the worst weeds spray directly on the weed. The steamer would be for areas close to the house. They are plug in so you need an good outdoor extension cord. Most last about 15-20 minutes before you need a refill. No chemicals, just good old fashioned water that heat up in the steamer. Direct it at the weed and basically you burn it with steam. No run off of chemicals, no worry for kids or pets and only kills the weed. With many of the weed killer sprays you may kill things you don't want and are harmful. Many claim not harmful to pets but I wouldn't want my pet in any kind of chemical. That is your other option buy a weed killer and a spreader that you push and it throws the pellets in the grass.
Boiling water, for thicker plants pour Epsom salt or ice cream salt or whatever your got at the base
I just had new grass put in. Weeds are taking over already, some of them are huge!
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See if these help.