The Shapiro delay was predicted in 1964 and observed by 1966, and is now a tool used to measure the mass of distant binary pulsars. The Terrell-Penrose rotation was published in 1959, but I can find no evidence for experimental confirmation. Wikipedia lists none, although it does reference a good animation. Everybody believes the math, but what physical proof is there? What would it take to observe something like the animation shows?
Well, there is aberration of starlight, which is basically the same effect, except the apparently rotating object is the observing apparatus rather than the source.
Because the speed of light is finite, it takes some time for light to travel through a telescope, during which time the telescope has presumably moved transverse to the line of sight.1 As such, the direction to point the telescope to center the star is slightly off from the "abstract spacetime diagram" representation of the pointing.
If you trace the light backward, the original source would only see the telescope aligned perfectly in the radial direction when the telescope is misaligned in the static clocks-and-rulers reference frame.
Note that this was detected in the 18th century.
1 This is caused, for instance, by the Earth's motion around the Sun, which will have a component transverse to the line of sight except in very special conditions. It could be caused by other motion, too, but to detect it you need a time-varying transverse velocity.