Be careful with succulents like purslane. They can re-root fairly readily from fractured parts of the plant, so mowing might not be your best bet. Any herbicide used would need to be spiked to get through their waxy leaves and I cannot personally think of any herbicide you could use that wouldn't smoke your turnips. Hopefully somebody else has a better idea but I'm afraid your best bet, unless your turnips will eventually outcompete, might be to kill the field and settle for a cereal grain. Good luck.