I have this stuff all through my lawn and garden. Normal weed killer had no effect on it. I want it gone.
This is a section of the lawn:
This what it does in the garden when it's not mowed:
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It looks to me like a type of sorrel, along the lines of
This PDF from davis has some detailed info on how to control two rumex species. This might be a good place to start.
That said, sorrel is quite tasty, so if it's one of the edible species, you could harvest it (though the weedkiller already applied might negate that option).
Its French sorrel- quite tasty, lemony if I remember- weed kill it(something systemic) with a touch of stump killer/brush wood killer- paint it on when young and sappy with a brush and wait three weeks- remember its got a very long tap roots so digging it out has to be done when the ground is soft think about 12 inches or more in length if you want to go for that option.