Aircraft carrier is NO FAST MOVING target. It cannot run away. The attack on carriers can come from aircrafts and enemy ships. Brahmos type missiles with long ranges are very deadly for carrier.
ACs don't rely on Mithun Chakraborthy type bullet dodging skills to survive. They have their own protection suite both below and above water. They have long range radars, they have anti submarine warfare units, they have other submarines defending them, plus they have fighter jets to intercept enemy aircraft from a distance.
It is not child's play to kill a carrier with a single strike. It can be done with larger missiles, but if the situation escalates to the point where any country sinks another country's AC, it would in turn signal to the other party that a full fledged war is on. If the US sinks Chinese carrier, China might initiate a cyber attack, or use their satellites to kill US satellites, or start dumping US dollars and create a mass panic in the US.
Killing someone's AC is a different level of escalation. It can lead to a nuclear war. These type of skinking games are alright to play with random submarines or a corvette. All major powers know this. Once an AC is commissioned, the other powers accept the fact that it is going to be around for an eternity and take note of the change in the power equation and assign a negotiated role to you in the global 'great game'.