The time when experimental evidence supporting relativity began to appear coincided with the rise of antisemitism in Germany following Germany's defeat in WWI.
Antisemitism was on the rise in Germany; Jews were being scapegoated for the country's defeat in the war. As both Jew and pacifist, Einstein was an obvious target. The complexity of relativity did not help either. Opponents such as Ernst Gehrcke and Philipp Lenard found it easy to cast doubt upon its labyrinthine mathematics.
The situation reached crisis point in 1921 when, paralysed by indecision, the Nobel Committee decided it was better not to award a prize at all than to give it to relativity. The arguments raged for another year until a compromise was reached.
At the suggestion of Carl Wilhelm Oseen, Einstein would receive the deferred 1921 prize, but not for relativity. He would be given it for his explanation of the photoelectric effect, a phenomenon in which electrons are emitted from a metal sheet only under certain illuminations. The work had been published back in 1905.
Thus, after considerable hand-wringing and even deferring the prize for a year, in 1922 the Swedish Academy made the decision to award Einstein the 1921 prize a year late, writing to Einstein on the 10th of November 1922 :
... the Royal Academy of Sciences has decided to award you last year's Nobel Prize for physics, in consideration of your work in theoretical physics and in particular your discovery of the law of the photoelectric effect, but without taking into account the value which will be accorded your relativity and gravitation theories after these are confirmed in the future.
(Source : Mass and Energy in Relativity Theory, by Lev Borisovi.)
At this stage relativity was open to future consideration. However, Einstein chose not to attend the ceremony to receive the award in person. This decision may have made a future prize for relativity more difficult, if not impossible, since it gave the impression that Einstein did not value the award.
Having said that, at the time Einstein received confirmation of the award from the Nobel committee, he was committed to a lecture tour of Japan.
German foreign minister Walther Rathenau had been murdered by anti-Semites. In the subsequent investigation, the police had found Einstein's name on a list of targets. In the face of such a death threat, leaving Germany to spend months in the Far East, rather than a few days in Stockholm, must have seemed prudent.