I hate to suggest chemical warfare, but that's a broad-leafed weed in a lawn, and you can get things that kill doctyledonous plants like that without harming monocotyledonous grasses.
The reason I hate to suggest it is that some of them are so persistent that you can use them to clear weeds out of a pasture, let horses graze in that pasture, heap the horsemuck in a corner, let it rot for a year, spread it around the roses and kill the roses because the weedkiller's still active after all that.
The alternative is extreme trial of patience and your knees: paint it with glyphosate everywhere it comes up. It's probably linked underground (easy to check by pulling one out and seeing whether a fat white rhizome comes with it) and if it is you won't have to hit every surface piece, just enough of them in each "grid square" of the lawn.
If you really want to do it all physically, I suppose you could cut two rows of turfs, then do a row at a time, cutting row 3, laying it on top of row 5, putting row 2 upside-down where row 1 was, cutting row 4, laying it on top of row 6, putting row 3 upside-down where row 2 was and so on. No guarantee of it killing things, but if you then turfed over that you'd be giving the new lawn a major headstart. Put some broad-leaf-only weedkiller between turned turfs and new turf and you've probably got it. Just be really careful not to get it on anything else.
Charlie November, these paragraphs would give me a headache, let alone if I was an inexperienced gardener!!