Turkish coffee is served with water for the same reason that a plate of sushi comes with a side of pickled ginger root.
When drinking your coffee, you sip your water first, then your coffee, and then your water again, your coffee, water, coffee, and so on. Between each sip of coffee, you drink water to cleanse the palate. The result is each sip tasting as good as the first.
The practice can be extended to other drinks, too. I sometimes do it when I drink soda. It works great.