First, China has one carrier, and it is not operational, just a testing hulk used to develop the plans, procedures, and capabilities that the US developed in the 1920s. Second, the US has a dozen aircraft carriers, and they're protected in a battle group by many picket ships, including attack submarines, frigates, destroyers, cruisers, as well as an entire air wing. Force projection means that each of those picket ships stays well away from the carrier, but keep her within their protection envelope. The subs and frigates are very adept at anti-submarine warfare (ASW), while destroyers do both that and anti-air warfare (AAW). Cruisers do a bang-up job of AAW. On top of that, you have sentry aircraft that stay hundreds of miles in front of the carrier, all the time, and are always monitoring for enemy ships, aircraft, and submarines. Once they've spotted something, the two (or more) aircraft flying interdiction screens can move to intercept.
China has a military that is still working on things that the US developed in the 1970s. The difference is that they've got a lot more people.