I looked up the word for "brother" in other Turkish languages. In Ubzek it is aka. And in Volga Tatar the corresponding word is abi. The word "kardesh" sounds suspiciously similar to the Armenian word kardash which means stone carver. So could this possibly be a loanword from one of the Indo-Iranian languages spoken in the region?
Old Turkish (from 8th century on) has kadaş and ka kadaş in the meaning “kinsman”. Anatolian Turkish kardeş results from a folk-etymological reinterpretation of the old word, as if from karın "womb" + daş "sharer". Clauson in his "Etymological Dictionary of pre-13th-century Turkish" argues that ka is a loan from Middle Chinese, and kadaş a derivative of the former. This may or may not be correct, but in any case the form with -r- is very late.