GPS satellite atomic clocks must be adjusted by about 38µs/day to stay in sync with Earth-bound clocks due to the effects described by general & special relativity.
The time slowing effect of a stronger gravitational field described in general relativity means a clock on Earth runs about 45µs/day slower than the clock in the satellite. While the time dilation effect described in special relativity, due to the relative motion, accounts for the satellite clocks falling further behind by about 7µs/day. The net effect is 45 - 7 = 38.
When discussing this in these terms of tiny numbers it might seem like so small of a difference as to be irrelevant to navigation but tiny errors add up. Without taking these effects into account your GPS would be next to useless in a matter of hours and off by about 10 kilometers by the end of the day.
Both effects have been experimentally confirmed in numerous different ways with a very high degree of accuracy, giving us an extremely high degree of confidence that the effects are real and not measurement errors and that the theory accurately describes the observed effects and that Classical Mechanics alone cannot account for the observed data.
GPS satellites travel at roughly 4000 m/s, you can compute the exact daily cumulative effect on time using this Wolfram Alpha calculator: