Well does not the same etymology also apply to Tur in Turk?
You mean Turan and Turk derive from the same root?
No. There are many theories where the name Turk derives from, it is heavily disputed but the etymology of Turan is certain.
Here is what wiki says for Türk:
''It is generally accepted that the name
Türk is ultimately derived from the
Old-Turkic migration-term
[59] 𐱅𐰇𐰼𐰰
Türük/
Törük,
[60] which means 'created, born'
[61] or 'strong',
[62] from the Old Turkic word root
*türi-/
töri- 'tribal root, (mythic) ancestry; take shape, to be born, be created, arise, spring up' and derived with the Old Turkic suffix 𐰰 (
-ik), perhaps from
Proto-Turkic *türi-k 'lineage, ancestry',
[60][63] (compare also the Proto-Turkic word root
*töre- to be born, originate').
[64] ''
Here what it says for Turan:
''In ancient Iranian mythology, Tūr or Turaj (
Tuzh in
Middle Persian)
[6] is the son of the emperor
Fereydun. According to the account in the
Shahnameh the nomadic tribes who inhabited these lands were ruled by Tūr. In that sense, the Turanians could be members of two Iranian peoples both descending from Fereydun, but with different geographical domains and often at war with each other.
[7][8] Turan, therefore, comprised five areas: the
Kopet Dag region, the
Atrek valley, parts of
Bactria,
Sogdia and
Margiana.
[9]
A later association of the original Turanians with
Turkic peoples is based primarily on the subsequent
Turkification of Central Asia, including the above areas.
[10][11] According to
C. E. Bosworth, however, there was no cultural relationship between the ancient Turkic cultures and the Turanians of the
Shahnameh.
[12]''
