It's kind of a principal of conduction that electron scattering, and power losses due to electrons scattering and bumping into atoms and vibrating them(atoms vibrating equals heat), increases proportionally with current density in a conductor. This is known as I2R losses in a conductor. The more current in through a smaller conductor, the more scattering and the more losses due to heat. So larger conductors are used to keep this loss to a minimum.
This scattering doesn't seem to happen in superconductors. So, in theory, a huge current could be sent through a small conductor without these losses. But there is also something know as electromigration, atoms actually being knocked out of the lattice structure due to being bombarded by too many electrons, which can cause degradation of the conductor material at super high current densities.