It starts off very shallow-rooted, red stems and eventually sprouts a grean leaf. Then, it spreads very similar to spurge, however the roots and stems become very thick, eventually sprawling over the entire lawn, sometimes more than a foot long. Please help in identification so we can figure out how to get rid of these suckers!

Thanks

However, from the pictures and description it looks like purslane, a relative of portulaca which people plant for flowers.

In a lawn, it is easily controlled with a broadleaf lawn care product, such as one containing 2,4-D, when it is small.

It takes advantage of bare areas to get going and doesn't do well when the lawn is thick, so working on whatever is causing you to have a thin lawn would eliminate it just as well as herbicides. Herbicides will give you a quick fix but have to be repeated at least annually, dealing with thin grass is the long term solution that will eliminate the need for herbicides.