No, time does not move faster if you travel west on Earth. The perception of time is not influenced by the direction of travel or the rotation of the Earth.
However, there is a phenomenon called time dilation in special relativity, which predicts that time can be affected by relative motion and gravitational fields. However, the effects of time dilation in everyday situations on Earth are extremely small and practically negligible.
In special relativity, time dilation occurs when an object moves at high speeds relative to an observer or when it is subjected to a strong gravitational field. For example, if someone were to travel at speeds close to the speed of light or be near a massive object like a black hole, significant time dilation effects would be observed.
However, for normal travel on Earth, the speeds involved are far below the speeds necessary to experience noticeable time dilation. The speed at which the Earth rotates and the speed at which we typically travel on its surface are much slower compared to the speed of light. Therefore, any time dilation effects associated with our everyday movements on Earth are too minuscule to be detected or experienced.
In practical terms, the perception of time remains consistent regardless of the direction of travel or the rotation of the Earth for ordinary terrestrial situations.