What is this plant?
Thanks for your question. It appears you have a very invasive weed on your hands. It's known as Bishop's weed, goutweed or ashweed and is very difficult to get rid of.
The plant spreads primarily by underground rhizomes. If you leave even a tiny piece of rhizome behind when digging it up, it will produce more plants. The more pieces, no matter how tiny, the more plants.
For relatively small areas, constant vigilance and diligence in removing it will eventually pay off.
Solarization is a possible option though it would mean losing your garden for a season. Cover the area with tarps or cardboard and weight down. At the end of the season remove the cardboard and replant.
You may eventually have to resort to some sort of herbicide, but before you go the poisonous route, you might want to try an iron-based alternative. Fiesta is just one.
With any herbicide be sure to follow the label directions and apply accordingly.