From the Latin turbinatus meaning top-shaped and the Greek carpos meaning body - a reference to the top-shaped growth of the stems.
I think there's an error in this description, and "carpos" does *not* mean body, but means "fruit". So it's top-shaped fruit rather than top-shaped body. Am I wrong?
No you are correct Armen, carpos means fruit, as in Ariocarpus:-
"Ariocarpus" is derived from the Greek Aria meaning "Sorb" (Sorbus aria – Rosaceae - is the white-beam tree) and the Greek word karpos meaning "fruit". The Genus name implies: "fruit similar in appearance to that of the white-beam"
Also Yavia cryptocarpa means "hidden fruit"
"crypto- before vowels crypt-, word-forming element meaning "secret" or "hidden"
In my book "What does that name mean?" (translated from Dutch), a Succulenta book aimed at Botanical names for succulents, it says it means it is shaped like a 'draaitol' in Dutch, shaped like a spinning top in English.
The explanation does not define it being the fruit or the plant body. But likely indeed it would refer to the fruit. But looking at the fruits I guess that name would be applicable to more succulents...