We report extensive radio and X-ray observations of SN 2003bg, whose spectroscopic evolution shows a transition from a broad-lined Type Ic to a hydrogen-rich Type II, and later to a typical hydrogen-poor Type Ibc. We show that the extraordinarily luminous radio emission is well described by a self-absorption-dominated synchrotron spectrum, while the observed X-ray emission at t-30 days is adequately fit by inverse Compton scattering of the optical photons off of the synchrotron-emitting electrons. Our radio model implies a subrelativistic ejecta velocity, -0.24c, at t sub(0)-10 days after the explosion, which emphasizes that broad optical absorption lines do not imply relativistic ejecta. We find that the total energy of the radio-emitting region evolves as E-7.3 x 10 super(48)(t/t sub(0)) super(0.4) ergs, assuming equipartition of energy between relativistic electrons and magnetic fields (e sub(e) = e sub(B) = 0.1). The circumstellar density is well described by a stellar wind profile, with modest (factor of 62) episodic density enhancements that produce abrupt achromatic flux variations. We estimate an average mass-loss rate of -3 x 10 super(-4) M sub( )yr super(-1) (assuming a wind velocity of u sub(w) = 10 super(3) km s super(-1)) for the progenitor, consistent with the observed values for Galactic Wolf-Rayet stars. Comparison with other events reveals that 650% of radio supernovae show similar short-timescale flux variations, attributable to circumstellar density irregularities. Specifically, the radio light curves of SN 2003bg are strikingly similar to those of the Type IIb SN 2001ig, suggestive of a common progenitor evolution for these two events. Based on the relative intensity of the inferred density enhancements, we conclude that the progenitors of SNe 2003bg and 2001ig experienced quasi-periodic mass-loss episodes just prior to the SN explosion. Finally, this study emphasizes that abrupt radio light-curve variations cannot be used as a reliable proxy for an engine-driven explosion, including off-axis gamma-ray bursts.