Flea beetle is likely to be the culprit on your turnips - turnips like radishes and swedes are part of the Brassica family and all are prone to attack by flea beetle especially seedlings which can often be wiped out quickly. If your leaves are riddled with holes and you see shiny black beetles which hop away when disturbed then this is definitely the flea beetle. The beetles emerge in April and May after winter hibernation, ready to start feeding. The eggs are laid near host plants, hatch in late summer and feed for a few weeks before hibernating. Due to the Indian summer we have been having it’s no surprise that you still have some present.
I would suggest spraying with our Organic Insect Killer Spray made from nature’s own quick acting insecticide pyrethrum. Take care not to knock the leaves or the beetles with jump off prior to spraying!