Learn Turkish: Adam kaba değil, çok kibar. - The man is not rude, he is very polite.

Yes, you absolutely can! You could say Adam kaba değil, ama çok kibar. However, just like in English, when you put a negative statement right next to a positive contrasting one, the meaning is obvious. Turkish speakers often just use a comma instead of ama to link them smoothly.

Where is the word for "is" in this sentence?

In Turkish, the "is" for "he", "she", or "it" is usually invisible! No suffix or extra word is needed. Adam kaba değil simply translates to "The man is not rude." The same happens in the second half: çok kibar translates as "is very polite."

Why do we use the word değil for "not" here, instead of a negative suffix like -ma or -me?

Because we are negating an adjective (kaba meaning "rude"), not a verb. Suffixes like -ma and -me are only attached to action verbs (like reading or running). To say that a person or thing is not a certain noun or adjective, you must use the separate word değil.

Why does the second part omit the word for "he"?

Turkish frequently drops pronouns like o (he/she/it) when it is already clear who the subject is. Since the first part of the sentence establishes we are talking about adam (the man), there is no need to repeat it or add a "he" in the second half.