Time dilation is a phenomenon predicted by Einstein's theory of relativity, and it is indeed caused by relative motion between observers. However, it is not limited to translational motion alone.
There are two main types of motion that can lead to time dilation: relative velocity (translational motion) and gravitational acceleration (curved spacetime). Both types of motion can cause time dilation, and they are interconnected through the principle of equivalence in general relativity.
When an observer moves at high speeds relative to another observer, time dilation occurs due to their relative velocity. This is known as "kinematic time dilation." As an object approaches the speed of light, time dilation becomes more pronounced. This effect has been observed and measured in various experiments, including with high-speed particles in particle accelerators.
In addition to kinematic time dilation, there is also "gravitational time dilation." According to general relativity, the presence of a gravitational field can cause time to flow differently. Clocks in stronger gravitational fields run slower than clocks in weaker gravitational fields. This effect has been observed and confirmed through experiments such as the famous Pound-Rebka experiment.
Now, coming to your specific question, if you were turning around a fixed point, you would indeed experience time dilation. This is because even rotational motion can be considered as a form of accelerated motion, and acceleration leads to time dilation. However, the effect would be relatively small unless you are rotating at extremely high speeds.
It's important to note that the magnitude of time dilation depends on the speed or acceleration involved. For everyday motions and velocities, the time dilation effects are negligible and not easily noticeable.