Once the soil is infested with those things, turnips and to a lesser degree radishes are toast. There is not much available to a home gardener at this time that will control them. Diazinon will to some extent but it is no longer available to home gardeners. Should not bother the beets. The greens only type turnips like Seven Top tolerate them much better than the root types.

Root maggots are the larvae of a small fly which lays eggs on the plants. When the eggs hatch, the larvae go into the soil and feed on the roots. Row cover will keep the flies off the plants. However, if larvae are already in the soil they may emerge under the row cover, in which case it won't work. I don't know how likely that is to happen. I have had success using row cover. It's worth a try.

Jim