Why don't PC power supplies require heavy transformers?
The fundamental reason why switching power supplies can use a much smaller transformer is because they use a frequency that is typically a thousand times higher than 50 or 60 Hz.

This means that the primary inductance (normally around 10 henries for a conventional AC transformer) can be reduced to circa 10 mH. It then follows that the core only needs to be wound with a fraction of the number of turns and it makes the core a lot smaller.