Note that the problem is not in visualizing the 4th dimension, that an easy thing to explain. The problem more related to why we are in 3D that moving along 1D time dimension?
You cannot avoid defining time in terms of change. In the same way that if there were no changes (dx/dy etc) in a terrain the map would be totally uniform and uninteresting, if the terrain did not change in time, time would be uniform and undefinable. Repetitive changes allow us to define time ( no need to go to entropy, the solar system, day/night etc are enough ) and clock it/ measure it.
Time projects into the 3 dimensional world. Geological strata ( and many other proxies) assign to each (x,y,z) a time value on the axis of t. Thus time can be projected into (x,y,z).In a similar way space dimensions project into time. The time taken to walk to the station has one to one correspondence with the distance in kilometers.
So time is a necessary dimension to describe the changes seen in three dimensions, in a similar way that a third space dimension is needed to describe the projections of a sphere to two dimensions.
Entropy must come in for a classical definition of the arrow of time and non reversibility. A rough discussion on disorder, broken glass not repairable etc should give them the concept.
Thus even without special relativity time can be thought as another dimension since it projects into the spatial ones. One can then go on to special relativity as surprising us with the different type of dimension ( pseudo euclidean) it turns out to be ,from experiments.
p.s. with this view of how time is defined in our experience we can also with assurance say that there is only one time dimension. If there were two time dimensions the functional dependence of changes in the space three dimensions would be complicated. It would be a many to one projection, similar to projecting a three space dimensional object to one space dimension.