Removing Common Mallow from New Lawn?

I have a yard that is 80% weeds. So I soaked the whole lawn with Roundup, then I rototilled the yard. I threw away 12 garbage bags full of weeds then I put carpet over the top for 2 months hoping to kill anything left.

I pulled off the carpet and leveled the yard with my rake. I put in a sprinkler system and a thin layer of good top soil (bought from the store). I planted grass seed, added turf builder fertilizer and after about 4 days grass started coming in. I wanted it thicker so I put more seeds down. Both times I raked in the seed to cover it with a little soil. I had to leave it for a week. When I came back to it and I find all these weeds called mallow common.

My next move is to cut down to 1 watering per day and maybe cut the grass for the first time. I live in northern California so it's still warm. How do I get these weeds out? Thank you.

By Frank