Learn Turkish: Şu garson bir Türk kahvesi içiyor. - That waiter is drinking a Turkish coffee.

This is an indefinite noun compound, which is how Turkish combines two nouns to make a new concept. The first word (Türk) stays exactly as it is, and the second word (kahve) gets a possessive suffix. Because kahve ends in a vowel, we add -si. Literally, it is built like 'Turk its-coffee'.

Why is bir placed before the whole phrase instead of in the middle?

The phrase Türk kahvesi acts as an unbreakable single unit. When you want to say 'a' or 'one' of these noun compounds, the word bir must go right before the entire block. You cannot split the compound by putting bir inside it.

Both şu and o mean 'that'. Why use şu here?

Şu is used for someone or something you can physically see and draw attention to right now, like pointing across a cafe. If you use o, it usually means the person is further away, or you are talking about them without necessarily pointing at them.

Since the coffee is the object being drunk, shouldn't it have the accusative case ending?

If the waiter were drinking a specific coffee ('the Turkish coffee'), we would add the buffer 'n' and the accusative suffix 'i' to make it Türk kahvesini. However, because he is drinking an unspecified coffee ('a Turkish coffee' marked by the word bir), the object stays in its default form without the accusative ending.