I don't know about changing the soil, but there is a lot of soil around the country that is deficient of sulfur. Plants never used to have a problem with all the pollution (sulfur dioxide) in the atmosphere. Plants could take it in through leaf tissue, that could/would satisfy the plants sulfur needs. Now that the country has cleaned up all the air pollution, it has left you no choice, but to add sulfur to your ground if you are deficient. Then the plants can take it up as a sulfate Ion. Soil test is the best way to find out how much Sulfur one would need to add for your specific crop. Turnips use about 40lbs of actual sulfur per acre. If your turnip leaves start turning red or have red streaks in them, good sign you are short of sulfur. The lack of sulfur can and will retard the protein synthesis that takes place inside your turnip plant and most other plants.
Maybe more the plant lacks in sulfur, the worse the taste? Maybe a combination of these things make the turnip less desirable?
Whip might know more about this subject. What you say whip?