"I need to drink water."
Translation:Su içmem lazım.
Lazım is always used with a noun in nominative. Like "Para lazım" (we need money / money is necessary).
When you have a verb, you have to turn into a noun -which by the way is an extremely simple thing to do: just omit the final K from the infinitive: içmek (to drink) -- içme (drinking).
From that point on, you can "possess" it like any other noun. Just like how you would say: "pencere" (window) -- pencerem (my window), you can also say: içme (Drinking) -- içmem (my drinking).
içmem lazım = my drinking is necessary = I need to drink.