If you are at all mathematical, you can verify that relativity affects GPS signals by calculating what difference both special relativity (satellite clock moving faster than clock on Earth, hence slower) and general relativity (satellite clock higher up the gravitational field than clock on Earth) would make to timekeeping and hence accuracy of location. The effects work against each other, but one is larger than the other.

You can verify accuracy of location of a GPS yourself. IME this is almost always considerably less accurate than published estimates by the device manufacturer, but still impressive. However, you need to be careful - most smartphones use multiple technologies to determine their location, not just GPS, so will be more accurate than the GPS signal can possibly be.