We have a very difficult to shift weed in our garden. It has long redish root, which is long and narrow that simply snap when you try to remove it. They seem to travel straight down rather than outwards. It has taken over our lawn.
I have not been able to find reference to it in any of the books I have looked at or on any of the web sites I have seen.

Hard to work out the scale from your photos, but my guess would be creeping buttercup. If I'm right, it kind of loops over the ground and then sends down roots, altho they're white in my experience.. If I'm wrong, someone will soon jump in. If it's only in your lawn, the easy answer is to apply a "weed and feed" dressing in the Spring.