I had those exact weeds in my new lawn ever since the new sod started growing. It took almost 6 weeks to get rid of them. I started by just pulling them by the root but again they would return the very next week. I started hand spraying each one with a normal weed killer right before I new it would be hot and sunny for a couple of days. I did not pull them and continued to spray until they had all dried up and died. Be sure not to water after you have sprayed the weed killer or it will wash away what you have done. I learned this the hard way! I am guessing you have bermuda cause thats what I have and I my new lawn was covered in these onion looking weeds. They especially like to grow in the corners and around the 2 trees that were planted. Just be patient and stay on themt and you can knock these out in no time! Hope this helps!!

Ok - I finally found out what this is - it's called mulberry weed.

Most likely this is Broussonetia. Second option is Morus alba (same family, Moraceae). Both have very variable shape of leaves, even on the same branch differently looking leaves can be found. Here is a link that might be useful: Broussonetia

Might be asparagus fern, A. densiflorus. Tubers on the root system suggest the same. hortster