Learn Turkish: Genç kadın bir şarkı söylüyor. - The young woman is singing a song.

This is because of a rule called vowel narrowing. The root of the verb is söyle-. Because it ends in a vowel (e), we can't just add -iyor directly. Instead, the final e changes into a narrow vowel. Due to vowel harmony with the ö in the root, the e changes to ü, giving us söylüyor.

I thought söylemek meant to say or to tell. Why is it translated as singing here?

You are right! By itself, söylemek means to say or to tell. However, in Turkish, you don't sing a song; you say it. The set phrase şarkı söylemek is the standard way to express singing a song.

Why is bir placed before şarkı and not at the very beginning of the sentence?

Word order changes the meaning here. Genç kadın (the young woman) is the subject of our sentence, and bir şarkı (a song) is the object. If you put bir at the very beginning (Bir genç kadın...), the sentence would mean A young woman is singing a song instead of The young woman.

Why doesn't şarkı have a case ending on it?

In Turkish, specific direct objects (like the song) get an accusative case ending, but indefinite objects (like a song) do not. Because she is singing bir şarkı (a song) and not the specific song, the word şarkı stays in its plain dictionary form.