The following example is something I came up with after reading an example given in the prologue of the book Einstein gravity in a nutshell by A. Zee

In his 1905 paper, Einstein starts by invoking two postulates:

1. The laws of physics are the same for all observers when moving at constant velocity relative to each other. This is the Galilean principle of relativity as it is known.
2. The speed of light c, is the same for all observers in any frame of reference.

This leads him to certain conclusions, which have been elaborated on and expanded throughout the last 100 odd years.

The loss of simultaneity is depicted as the 3rd example in the prologue of Zee’s book

He describes a situation not unlike the one below in order to illustrate the loss of simultaneity.

The following is a slightly modified version of the example given in the book, which (I hope) illustrates time dilation, and the constancy of the speed of light.

The situation to be visualized is as follows: You are moving relative to the ground, say on a train. You witness an arrangement on the ground passing by you (which has been elaborated to you beforehand, so you expect to see something). The arrangement involves two beams of light fired at a screen from opposite directions, and equidistant from the screen. Consider that a loud sound is generated if the beams arrive at the screen at exactly the same time. Also, consider that if any one beam hits the screen, it immediately changes color from white to green and if both the beams hit the screen at the same time, it immediately changes color to red. Now, the arrangement is such that it is agreed that the beams will be turned on at exactly the same time as measured by a clock next to its respective switch and the clocks will stop at the exact instant the beam is turned on.

Your train is passing this arrangement and you hear a noise indicating that the beams have hit the screen, apparently at the same time.

You stop and consider the situation.

For the two people manning the switches on the ground, it is perfectly clear that they have turned on their respective switches at exactly the time prescribed by a clock near the switch. It is also clear that the clocks, being on the ground, can be considered stationary, with respect to the moving train which also carries with it a similar clock. Also, the clocks have been previously synchronized just before the setup, and they are accurate enough to not lose track within a noticeable period. Thus, the operators will see the beam hit the screen, the time displayed by their respective clocks will be the same, and the sound generated is heard by them both at exactly the same time. The color of the screen should be red as seen by the operators.

For yourself, moving along with the train at a speed vt, relative to the ground, you hear the sound all right, but visually, by an ingenious mechanism on the roof of the train, the onboard clock is triggered to stop at the agreed prior moment and take a snapshot of the screen as it passes by on the ground. The mechanism also records the color of the screen as white. You check the time on the onboard clock, and find it stopped at the agreed time. You think, how can this be? The color of the screen is white, indicating that the beams did not hit the screen at the agreed time. Also, the time registered on your onboard clock is in agreement with the prior arrangement. In this case, you would conclude that indeed the beams may have been triggered, but later than the time registered on your onboard clock. You hop off at the next station and make your way back to the arrangement. You find that the color of the screen is now red.

Now, you think, I was right about these imbeciles, they did not turn on the beams when they were supposed to! There has been a delay. You approach them and talk. The operators, for their part, vehemently deny any delay from their end and request you to inspect the time on each of their clocks which you proceed to do so. You find the times on both the clocks to be in agreement with the prior arrangement. So, you pay them and see them off, and wonder about this discrepancy.

These are the facts you have: The two beams each have been turned on at the agreed time. The color of the screen is red indicating that the beams did hit the screen at the same time. The color of the screen in your snapshot taken from the moving train is white, indicating that the beams had not yet arrived at the detector at the instant the snapshot was taken. The clock onboard the train did stop at the agreed time, so the snapshot was taken at the agreed time.

The only reasonable conclusion you can draw from this is that time must have been passing faster for you onboard the moving train. You clicked the snapshot just a tad bit earlier than the time as agreed and was noted by the stationary clocks on the ground. The onboard clock has also registered the agreed upon time, but the ‘time’ that it was measuring must have been passing faster than the time as measured by the stationary clocks.

So, what do you conclude? — Suppose you do not know for a fact that the speed of light is constant. Given your insight into ‘time’ now, you try to reason how this can be so.

Since you have deduced that time as measured inside your moving train was faster than the time as measured upon the stationary ground, you run through the sequence of events again.

The operators turn the beams on at the exact same time, relative to ground. You take a snapshot of the screen at the prior agreed time, from onboard the moving train, and the color of the screen in the snapshot is white. The beams reach the detector at the exact same time, relative to ground, and the screen becomes red. You were inclined to initially say that the operators delayed the timing of the start of the beam. Then, after examining the clocks on the ground, you accept that you made a mistake and note that the beams have indeed been started at the same time, relative to ground.

Thus, you deduced that time was flowing at a faster rate in the moving train.

Then you wonder why the snapshot taken from the train is white and not either green or red, if the ‘time’ thing you just deduced seems too fantastic an explanation.

You first note that the screen in front of you is indeed red in color, and not green. This must mean the beams arrived at the detector at the same time. And yet the snapshot shows white. This could only be if:

1. The beams had not reached the detector at the screen at the time the snapshot was taken. Which would be true as the snapshot was white, indicating that light takes some finite time to reach the detector.
2. The beams in fact did reach the screen and turn it red, but since the snapshot was taken at the instant of agreed time, the red light from the screen had not yet made it to the train which was some horizontal distance away from the arrangement. So, you ended up picking up the image of the screen as it was just a tad bit earlier than its actual state at that instant.

So, now you consider the following two possibilities:

1. Light travels instantaneously in air. In this case, the snapshot would have been red, which it was not. It was white.
2. Light travels at a finite speed in air. Thus, you are able to establish the validity of this statement.

Then, you think whether the finite speed of light is different for each of the two sources. You think, the screen is moving towards one of the beams of light, and away from the other. The beam that is approached by the screen should hit the screen first, and the beam that is approaching the receding screen will hit it some tiny fraction of a second later. This difference in timing would be exacerbated if your train were to be moving at ever faster speeds relative to the ground. But, the color of the screen is red, and a sound was heard by all parties, indicating that the beams did hit the screen at the same time. So, you think, what if light was travelling at different speeds? The beam moving towards the approaching screen, would be moving in the same direction as the train, and hence it would appear to be moving slower than its usual speed. The beam moving towards the receding screen would be moving in the opposite direction to the train and would appear to be moving faster than its usual speed. Since, you know that the three things in motion in this scenario are actually the two beams and the train, the screen must have been stationary. Now, as the screen isn’t moving, if the light beams were indeed moving at different speeds, the screen should have registered a green color, but the screen is red. You deduce that light was moving at the same speed, in both directions, and hit the screen as expected. You take another look at the screen, and find it to be red. Thus, you conclude that even though, from your perspective on the train, it appears at first glance that the light beams should have been moving at different speeds, they clearly hadn’t been doing so, and were moving at the same speed.

Thus, you conclude that light travels at some speed, which does not depend on the relative motion of an observer.

Next, you consider what would be the case if in fact, the snapshot taken from the train was indeed red as well. This would imply an instantaneous speed for light. There would not have been any discrepancy noticed by you or the operators regarding the experimental arrangement and the effects produced. Thus, you realize, that unless some experiment establishes the finite nature of the speed of light, the above thought process and the conclusions drawn are moot.

Luckily, you have the benefit of hindsight, and you know for a fact that the finite nature of the speed of light is widely accepted in the scientific community.

You proceed to summarize your conclusions.

You now have these facts, which have been arrived at in a way each not requiring the other:

1. Time moves slower for an observer in motion with respect to a stationary observer.
2. Light takes some finite time to travel between two points. This implies some speed of light exists, call it c.
3. The speed of light does not depend on the relative motion of an observer with a stationary observer.
4. It seems a bit of a stretch to deduce that the speed of light is constant across the universe just from these above three facts, but further analysis could be applied in a similar manner to do this. The motivations that led Einstein to use this fact as a starting postulate may shed some light on the matter (check this). But we note the remarkable consequences of making that assumption, which have been experimentally verified.