Learn Turkish: Hiçbir tehlikeden kaçınmadı.

The verb kaçınmak (to avoid / to refrain) requires the ablative case (-den / -dan). In Turkish, you don't just avoid a thing; you avoid from it. So, "to avoid danger" becomes tehlikeden kaçınmak.

Is kaçınmak related to the verb kaçmak (to run away)?

Yes! Kaçmak means "to run away" or "to escape." Adding the -ın suffix creates the new verb kaçınmak, which means "to avoid" or "to refrain from." Notice that both of these verbs require the ablative (-den / -dan) case because they involve keeping away from a source.

Why does the sentence use a negative verb (kaçınmadı) if hiçbir already makes it negative?

Turkish requires double negatives! Negative words like hiçbir (no/not any) or hiç kimse (nobody) must always be paired with a negative verb. So, hiçbir tehlikeden kaçınmadı literally means "he didn't avoid no danger," which we translate into proper English as "he did not avoid any danger."

Could I say hiçbir tehlikelerden to mean "any dangers" (plural)?