i checked the wiki subarticle "Consonants" and there is an example of /kar/ vs /ca:r/ (youglish link, as evidence for ":") which might not be a good minimal pair. do you know one? /g/ vs /ɟ/ would also work for me.
":" ruins it for me, because I consideer vowel length contrastive. "Karanın kası" means "the muscle of the land", but if I pronounce it as /ka:sı/, it sounds as if someone is dictating the letter k, giving "land" as an example word with initial k.
Using this web page as an informant, the example [kar] "snow" vs. [car] "profit" is a minimal pair. The only phonetic difference in the words is initial consonant: the vowels are not different in length (performances by the same speaker). In the context __C{#,C}, a long vowel is regularly shortened. It is true that the underlying forms differ w.r.t vowel length, but "minimal pair" is a property of phonetic forms, not underlying forms. Another example is [kap] "container, [cap] "ankles" (glosses are guesswork since that webpage is in Turkish). If you search kâ on the page it gives you all kâ-initial words (kâm retains a long vowel, surprisingly, which is useful for calibrating judgement of vowel length in the [car/kar] example).
