Modern aircraft carriers have reverse-osmosis desalination systems on board. These can supply hundreds (thousands?) of tons of potable water per day.

Yes, fresh water is a valuable commodity at sea; on U.S. Navy ships, water-flushed toilets are flushed with sea water, not fresh — or at least that's the way it used to be, and I can't imagine it'd be different today. (Source: Three years' sea duty as an engineering officer in the aircraft carrier USS Enterprise, CVN-65.)

EDIT: See https://www.navy.mil/submit/display.asp?story_id=103122 for a 2017 press release describing how the Reactor Department produces fresh water aboard the aircraft carrier USS Abraham Lincoln (a Nimitz-class carrier, i.e., an old design). TL;DR: It's done with giant evaporators, that is, distilling units.

Fun fact: At sea, one of the big uses of fresh water aboard a carrier is the repeated washing down of the aircraft to reduce corrosion from the salt air.