We calculate the structure of a relativistic shock wave in which the internal energy of the shocked fluid is radiated away on a timescale much shorter than the characteristic shock propagation time. The shock is assumed to move through a uniformly magnetized, neutral plasma consisting of protons, electrons, and positrons, and allowance is made for the possible production of electron/positron pairs in the shock itself. The radiation mechanism is taken to be synchrotron and inverse-Compton emission (the latter involving both synchrotron-produced and externally supplied seed photons) by the electrons and positrons. We simplify the discussion by considering a shock in which the magnetic field is transverse to the direction of propagation and focus attention on the properties of the radiative zone that forms behind the shock transition. In particular, we investigate the possibility that the compression induced by the cooling of the gas amplifies the magnetic pressure until it reaches (and ultimately exceeds) equipartition with the thermal pressure (which, in turn, limits the overall compression). We show that, if a significant fraction of the postshock thermal energy is deposited in the electron/positron component, then a considerable portion of the emitted radiation will come from regions of strong magnetic field even if the field immediately behind the shock transition is well below equipartition. This field amplification mechanism may be relevant to the production of synchrotron flares in blazars, miniquasars, and gamma-ray burst sources. We consider the latter application in some detail and show that this process may play a role in the prompt gamma-ray and possibly also the optical "flash" and radio "flare" emission, but probably not in the afterglow.