When you are accelerating upwards in an elevator, you feel an increase in apparent weight.
A scale measures the total amount of force pushing down on it, and if you were to step on a scale in an elevator, you would see an increase in its reading.
However, when accelerating upwards only the normal force increases — how does this translate to an increase in the "downward push" needed for an increase in apparent weight?
Similarly, when accelerating downwards, the normal force decreases but how does this translate to a decrease in the "downward push" needed to result in a decrease of apparent weight.
Best Answer
As you said correctly: as you accelerate upwards, the normal force acting on you will increase. Now look at the situation from the scale's "perspective" -- consider the forces acting on the scale:
Since the ground (i.e. the scale on which you stand) applies a normal force $F_N$ (upwards) on you, Newton's 3rd law says that you must also be applying a force $F_N$ (downwards) on the scale.
To summarize: the scale will measure the magnitude of the normal force acting on you; which is also the magnitude of the force you exert on the scale (by Newton's 3rd law).