why is the force on the side of a very narrow tank independent of the volume of water?

What you observed is known as the Hydrostatic Paradox.

You are confused that the force exerted onto the tank wall corresponds to the weight that is larger than the weight of water contained in the tank. You are probably familiar with levers: you can put a small weight onto the long arm of the lever and thereby exert a large force at the short arm. The gain in force is compensated however by proportionally longer path of the longer arm.

In your example, the water acts as such lever. By the way, this is how Hydraulic machinery works. The difference between the narrow and the wide tank becomes apparent if the lateral wall actually moves by, say, 1 inch. If the tank was 1 inch wide, the force reduce by half. If the tank was much wider, the force will remain the same and probably push the wall further, so the amount of potential energy stored in a wide tank is larger.

The point where this pressure law brakes down is when the surface tension of the water starts to play a leading role. You can try this yourself with a narrow tube (I would say make it considerably smaller in inner diameter than a sixteenth of an inch). If you submerge the tube in water and you pull it out, the molecular forces between the water molecules and the tube will keep some water in the tube, even if it is open at the bottom. The thinner the tube, the taller the column of liquid that can be held by the tube.

Spongy materials work the same way, by keeping small volumes of water in their pores. With very fine pores one can make a fairly large volumes of spongy materials that can hold plenty of water without the weight of the water being enough to drain it. If you were to fill a tank with such a sponge, the pressure on the sides of the tank would be partially compensated by the capillary forces of the sponge. The pressure on the bottom, of course, would remain the same as if the same amount of water were in there as without the sponge (plus the weight of the sponge itself). The downside of such a storage method is, that while the walls of the vessels could be made thinner, it would require extra pressure to squeeze the liquid out of the sponge.