In Einstein's famous thought example involving a fast-moving train (say with a velocity of $c/2$), we assume that lightning strikes occur simultaneously at the front and rear end of the train as viewed from an observer at a platform outside of the moving train. For an observer sitting at the center of the train, however, one of the strikes will then appear to happen before the other as the lightning strike which occurs at the front of the train will have a shorter distance to travel before it "catches up" with the observer. This I fully understand.
Now on to the part which confuses me. For the observer on the train, the two observed lightning strikes may be registered as two separate events, say event A and event B. The observer on the train could then, hypothetically, use a clock to estimate the time difference between the two events. For the observer on the platform, however, due to time dilation effects and the Lorentz factor, I would assume that a clock used by this external observer would register the time between the two events as longer than what the clock used by the observer on the train would show.
All this makes me a bit confused. The observer on the platform views the two lightning strikes as occuring simultaneously, but, at the same time, a clock used by this observer would show that longer time would pass between events A and B compared to the time registered by the observer on the train. This seems paradoxical to me, so if someone could clear up my confusion, then I would be extremely grateful!