In a web map, when you click on the center of overlapping points (that have the same xy), all the pop-ups are displayed (see 4pop.png).
If you click on the edge of the points (or just outside), only the pop-up for the top feature is displayed (see 1pop.png).
If you make the point symbol larger, the effect is exaggerated. You can click in the point symbol and get only the pop-up for the top feature (see BigSymbol.png).
This is a known behavior or limitation of the application. Fixing it is not in the current product plan.
I assume that it is not being address for performance reasons. That is why I suggest that it be an option that we be able to turn on in the settings of the web map.