Learn Turkish: Hasta adamla kaderdaşız. - The sick man and I are companions in fate.

The suffix -daş (or -taş) attaches to nouns to indicate fellowship, association, or sharing something. Here, it attaches to kader (fate or destiny). Therefore, a kaderdaş is someone who shares the same fate as you. The -ız at the end is the "we" personal suffix, making it "we are companions in fate."

Since kader has front vowels (a and e), shouldn't vowel harmony make it kaderdeş instead of kaderdaş?

You would normally expect that, but the -daş suffix is a bit of a rebel! While it sometimes changes to -deş or -teş (like in sesteş or özdeş), it frequently ignores vowel harmony entirely. It stays as a "flat a" in many common words regardless of the preceding vowels, such as meslektaş (colleague), dindaş (coreligionist), and kaderdaş.

The English translation says "The sick man and I", but there is no ben (I) or ve (and) in the Turkish sentence. Why?

Turkish very often uses the structure "[Person] + with ... we are" to express "[Person] and I". Here, adamla means "with the man". Paired with the "we" ending on kaderdaşız, it literally reads "With the sick man, we are companions in fate." This feels much more natural in conversational Turkish than saying Hasta adam ve ben...