Yes, if you were to move back and forth very fast, you would indeed experience time dilation. According to the theory of special relativity, time dilation occurs for objects that are moving relative to an observer at rest.
When you move at high speeds relative to a stationary observer, time will appear to pass more slowly for you compared to the observer. This means that clocks on your moving frame of reference would tick slower than those on the stationary frame. This effect is known as time dilation.
However, it's important to note that the amount of time dilation you would experience depends on your velocity relative to the observer. For everyday speeds that we encounter, the effects of time dilation are negligible and not noticeable. To observe significant time dilation, you would need to reach speeds that are a significant fraction of the speed of light, which is currently beyond our technological capabilities.
Nevertheless, the principle of time dilation is well established and has been verified through various experiments, such as comparing clocks on fast-moving particles with stationary clocks. So, while you wouldn't experience noticeable time dilation in your daily movements, it is a real and fundamental aspect of the theory of relativity.