The weed spreads out like a vine below the grass in the lawn and has a long root and pink flowers.
Edit: Here is a picture with the grass with the weed in it. You can see it at the bottom of the image.
I'm in South East, Queensland
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Chamaesyce maculata or Euphorbia supina. Hope this helps. Where is it that you live, what zone? I am guessing you have warm season grasses? Knowing the proper maintenance of the crop you are growing for a lawn will help so very much to suppress any weeds. One should not have to use herbicide, really ever and I am a 3 decade long licensed commercial pesticide applicator Missing the spot on these leaves as well so wait for others to help out as well.
Please send a picture of your entire lawn, the zone you live in, your maintenance practices, how often you fertilize, how short you mow your crop, what your watering practices are?
Here is another possible ID. Prostrate Knotweed...the distinctive difference would be the Spurge will have a milky sap. Do not get on skin. If no milky sap then not a spurge. Prostrate Knotweed