I have on my own turnips. I usually peeled it and didn't worry about it (home garden). If the damage went 'into' the turnip, then wireworms would be a likely source. Also noted as causing damage on roots of turnip are root maggots, but the damage from them should be more significant. It does seem like insect damage, and flea beetles have a grub stage, leading me to wonder if it might be the grubs of flea beetles. Anyway, this link has a list of the more common problems, but doesn't have photos of the damage to turnips. It is from Chrome browser.

https://www.plantvillage.com/en/topics/turnip/infos/diseases_and_pests_description_uses_propagation

Grow the turnips in a different are of the garden next year. For wireworm and root maggots, there aren't any chemicals available to home gardeners for their control anyway. If you had lots of little holes in the leaves of the turnips, that is flea beetle damage and you can control them. The active ingredient permethrin is very effective but not a natural or organic product.

Email me direct if you have further questions/discussion on this.---Jim