Answer: It sounds like your garden has been invaded by bindweed, a notoriously difficult plant to eradicate. Although in many cases repeated pulling will exhaust a weed, in my experience this is not one of them. You might try applications of glyphosate being very careful to follow the label instructions because this herbicide will kill desirable plants as well as weeds. To prevent additional seedlings from popping up you might also make sure to maintain a layer of mulch several inches deep throughout the garden. Good luck with the battle!
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