As with any perennial weed or undesirable plant, your options are physical removal or chemical removal. Physical removal entails either digging the plants up or covering them with a light-blocking tarp to starve them of sunlight. The goal of smothering unwanted plants is to deny them light for long enough that they run out of root energy stores, though it's hard to predict how long that will take; it might be several weeks.

Chemical options center around systemic herbicide use, which is absorbed by the plant (through foliage, in this case) and moved into roots where it kills the whole plant. Even so, more than one application might be needed before treatment is successful, and these kinds of usage directions (like how long to wait between sprays) will be on the product label. Note that these kinds of herbicides tend to be non-selective, so may injure or kill any other desirable plant contacted by the spray, so make applications carefully. County law might restrict use of non-organic herbicides, though, so you may need to clarify with county government if this use (to clear land and not selectively treat weeds in a lawn you're keeping) is permitted under their restrictions, which will otherwise prohibit the use of synthetic herbicides (which includes those that kill roots and not just top growth).

You may want to advise the person that weeds can still grow up through artificial turf; we've seen this happen, though don't know the details about whether the site was prepared well before laying the artificial turf or whether the artificial turf was aging and more prone to weed seed infiltration to the soil below.

Miri