Both are acceptable I believe, but they of course mean slightly different things.
In (1) it is the whole AP "new Tina Turner bob" which is modified by curly, and it means something like she has a new Tina Turner bob which is curly.
In (2) the AP "curly Tina Turner bob" is modified by new, this pushes the semantics in the direction that curliness is a property of the Tina Turner bob which is new to her.
How (2) is different becomes more apparent if you consider that there might be two or more types of Tina Turner bob, e.g. one with straightened hair, one with curly hair and one with fringed hair, which she maybe wore at different periods in time. Now if you want to distinguish these, you can only use the structure in (2) and refer to "her new curly/straight/fringed Tina Turner bob", but structure (1) doesn't offer you the same contrast, it's always a single prototypical Tina Turner bob which in her particular case might have turned out curly/straight/fringed/etc.
If you want to know more about this, you can look for papers on adjective scope, or for restrictions on multiple attributive adjectives in English, e.g. Bob Truswell's MPhil dissertation on that topic.
Edit to answer Keren's comment suggesting that the meaning explained under (2) would require the structure in (3): "Newly curly Tina Turner bob".
"Newly curly" has a different meaning and syntactic structure to both (1) and (2) above. While it is correct that adjectives are modified by adverbs, what I suggest in (2) above is not a structure where the adjective "curly" is modified. I suggest that the entire nominal phrase "curly Tina Turner bob" is modified.
The difference becomes apparent if we apply bracketing to mark out the syntactic constituents.
In (1), this gives us [curly [new [[Tina Turner] [bob]]]. As is apparent from this the entire phrase [new Tina Turner bob] is modified by "curly".
In (2), we have the analogue [new [curly [[Tina Turner] [bob]]]. As is apparent, the phrase [curly Tina Turner bob] is modified by "new".
In (3), we have [[newly [curly]] [[Tina Turner] [bob]]]. As is apparent from this bracketing, "curly" is modified by "newly", and the entire phrase "newly curly" is what now modifies the nominal "Tina Turner bob".
(3) is accordingly assigned a reading such as there exists a Tina Turner bob, which has acquired the new property of curliness; very different to both the structures the original post asked about.