Turkmenistan of all places is currently undergoing a gradual switch to Latin, which has happened in fits and starts with various mandates to do so, and then to stop doing so. But the culture seems pretty undeniably to be trending to romanization.
What do you mean? It's a set of symbols. No different from any other set, except potentially in size. What would a "natural fit" between a script and a language look like?
Arabic script? Turkic languages have lots of vowels, which are very important, and a lot less fricative consonants. You must add a lot of "diacritics" (points above or below).
Latin adapted for Turkish has same issues: a lot of diacritics, and also the infamous i/I distinction (like peek & pick in English) that causes typos.
Cyrillic for Turkic languages simply introduced more letters. For example, in Kazakh the vowels а, о, у, э, е, ы were extended with ә, ө, ұ, ү, і. Consonants got their fricative counterparts: к, г => қ, ғ. The only inconvenience I see is Latin "h" (e.g. жиhаз, furniture) which can't be capitalized -- it will collide with Cyrirllic Н (N). I actually find it easier to transcribe German in Kazakh letters than in pure Russian ones.
The flip side is you need a bigger keyboard, the number row gets used for letters too.
In Kazakhstan, the transition to Latin lost traction even in the state organs: every legal document is written in Cyrillic. In Uzbekistan the attempt was made ~25 years ago, IIRC, and still even state organs print a lot in Cyrillic. They allowed writing in both systems as legally equal for 10 years, but had to extend the transition period for another 20, IIRC.
So supposing it's the cultural trend is quite disputable -- I'd say only nationalists with decolonization agenda may want it (but then why not the old Arabic writing system?). The majority doesn't care nor want to switch.
In Europe, Polish and Irish are two examples of a language with extensive contrastive palatalization which uses the Latin alphabet, and they're probably among the least readable/elegant writing systems in Europe.
Sure, many languages have palatalized consonants, influenced by surrounding phonetic context (such as, vowel harmony in Turkic makes surrounding consonants "softer" too).
The true "contrastive palatalization" I was referring to is about having minimal pairs where words have totally different meanings based on whether the consonants are palatalized or not.
For example, in Russian (where I specify the palatalization of a preceding consonant with ʲ):
mer "a mayor"
mʲer "of measures"
mʲerʲ "measure!" (imperative)