Why put more than one winding on the low voltage side of a transformer?

Any given magnetic core of a certain area at a certain frequency will only generate so many volts per turn.

For instance, an iron core 12x25mm cross section is only good for about 0.1v rms per turn at 50Hz, needing 2400 turns for 240v mains.

You could in theory make a core with enough cross section to give you any number of volts per turn, but that would make for a very expensive and heavy transformer. It's cheaper and lighter to use less core area, and more turns, when making a small transformer, to match the voltages you want to use.

The inductance of the primary in a power transformer is the main factor that sets the 'magnetizing current', which is the current that flows with the secondary open. It needs to produce an appropriate level of flux in the core (not too low or the transformer will be too big and expensive, and not too high or the core will saturate).

You could not get enough inductance from a turn or two to keep the magnetizing current at mains frequency low enough for a small VA transformer with any core material unless the loops (and thus core volume and mass and thus cost) were enormous.

The turns required is thus dependent on the core permeability, saturation characteristics, and physical arrangement, the frequency and the voltage (or, equivalently, current) for a given VA transformer.

At high frequency and/or low voltage, you can indeed use a turn or two for a small transformer with an appropriate core for the frequency. Even at 60 Hz the high current secondary of a soldering gun transformer is only a few turns, and a (high frequency) switch-mode transformer may have a primary in the tens of turns rather than hundreds.