Answer: To some extent, taste will depend on the specific varieties you are growing and the growing conditions and weather. All three of these grow best if planted very early (before the last frost) in rich, moist soil and then grown quickly without interruption. In this case it sounds as though the dry weather may have contributed, keep in mind that a raised bed will dry out faster than the native soil.
Be sure to thin as needed, beets in particular need to be thinned to allow ample room for the beet to develop underground. Turnips tend to be bitter if grown during hot weather. (They grow best at 40 to 60 degrees so are usually grown for fall harvest rather than planted in spring.) They will be woody if the soil becomes dry while they are developing.
Beets will also become woody if the soil is allowed to dry out, but are more tolerant of warm weather. The beets should get sweeter as they size up.
Radishes will be hot or bitter if grown in dry soil or in hot weather. Hairy roots can indicate they are overgrown, meaning too old. Check the days to maturity on the seed packet -- some are ready very quickly so they are usually planted in succession, maybe a batch every week for a month or so to extend the harvest. Taste and shape depend on variety, some are peppery and some are sweeter, some are round, some cylindrical and some are actually supposed to be long and skinny.
Lack of size can also be caused by a shortage of potassium, so make sure you are using a balanced fertilizer such as 10-10-10 or, for root crops, 5-10-5 since they do not need a high amount of nitrogen. (If the green tops are luxuriant, but the roots are small, there may be excess nitrogen. This can happen easily if you used a lot of manure or excess fertilizer when preparing the soil.)
It is a good idea to run some basic soil tests and check both the fertility and also the pH. Your local county extension should be able to help with the tests and interpreting the results. This is especially important to do with a raised bed where the soil is a mixture of different ingredients.
Your daughter may have had a specialty radish, perhaps something like Parat Ginat Red. There are many different types available, especially if you look at specialty or gourmet vegetable seed suppliers.
I hope this helps.
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