Because "havuz" is in the nominative case, and not in the genitive case. In order to be "the pool's towels," "havuz" needs to be possessive, so it has to have a "-un" ending: "havuzun havluları."
When the first noun in the nominative case (not the genitive) and there's a possessive ending on the second noun, it's just a compound noun: "pool towels."
It's the Turkish equivalent of a compound word. If you want to specify that they're the pool towels (you're basically using "pool" as an adjective to describe "towels"), you leave the first noun alone, and put a possessive ending on the second one. There's more here: https://www.duolingo.com/comment/8616301
Exactly. (I hadn't had to figure out "dog bed" in Turkish, yet, so I asked Google to find me a köpek yatağı, to double check. And lo, and behold: https://www.google.com/?gws_rd=ssl#q=k%C3%B6pek+yata%C4%9F%C4%B1 =) )