These are some of the experiments which confirm time dilation. The story is that one of a pair of twins leaves on a high speed space journey during which he travels at a large fraction of the speed of light while the other remains on the Earth. Because of time dilation, time is running more slowly in the spacecraft as seen by the earthbound twin and the traveling twin will find that the earthbound twin will be older upon return from the journey. The basic question about whether time dilation is real is settled by the muon experiment . The clear implication is that the traveling twin would indeed be younger, but the scenario is complicated by the fact that the traveling twin must be accelerated up to traveling speed, turned around, and decelerated again upon return to Earth.
Accelerations are outside the realm of special relativity and require general relativity .
Despite the experimental difficulties, an experiment on a commercial airline confirms the existence of a time difference between ground observers and a reference frame moving with respect to them. One of the great successes of the quantum theory was the prediction of the energy levels of the hydrogen atom. When attempts were made to explain the fine structure of the hydrogen spectral lines, it was found that the splitting of the lines was in error by a factor of two. It was realized that relativistic time dilation must be used in calculating the frequencies, and calculations showed that this relativistic correction, called Thomas precession, was the factor of two which was needed for agreement with experiment. A precise measurement of time dilation was made in a double-photon experiment by Kaviola et al. in 1985.