So why would the observers moving with the moving rod find the two clocks not to be synchronous? Does light travel at different speeds in a moving frame, slower in the direction of motion and faster opposite the direction of motion? That's exactly what Einstein said. I know others can't speak for Einstein but everyone who has been posting here seems to support his theories so maybe you would like to help him out by explaining how that makes sense?
We imagine further that with each clock there is a moving observer, and that these observers apply to both clocks the criterion established in § 1 for the synchronization of two clocks. Let a ray of light depart from A at the time tA, let it be reﬂected at B at the time tB, and reach A again at the time tA′. Taking into consideration the principle of the constancy of the velocity of light we ﬁnd that
tB − tA = rAB/c − v
t′A − tB = rAB/c + v
where rAB denotes the length of the moving rod—measured in the stationary system. Observers moving with the moving rod would thus ﬁnd that the two clocks were not synchronous, while observers in the stationary system would declare the clocks to be synchronous.