Thin current sheets with half-widths in the range of about 10 or less ion inertial lengths (or ion gyroradii) have been identified as the sites of important dynamical phenomena in space plasmas. Recent space observations established that thin current sheets often have a bifurcated (double-peaked) current density. Earlier suggestions of possible bifurcation mechanisms are based on the presence of microfluctuations, magnetic reconnection, or a normal magnetic field component or assumed simplified models of adiabatic dynamics. Despite these efforts, the cause (or causes) of the formation of thin bifurcated current sheets and the conditions under which they form have remained unclear. In this paper, we identify quasisteady compression of a plane, initially wide collisionless current sheet as an effective physical mechanism for the formation of thin bifurcated current sheets. Our main tool is electromagnetic particle simulation. The initial sheet has a singly peaked current and a half-width that is five times larger than the ion inertial length. This sheet is quasisteadily compressed by external forces. A three-scale structure develops and the current bifurcates during compression. It is shown that the bifurcation, pressure anisotropy, and other major properties of the embedded current sheet can be understood in terms of basic physical principles, such as electric field shielding and momentum conservation. Sufficient conditions for bifurcation of symmetric current sheets are presented. They suggest that bifurcation must generally occur by quasisteady compression if the (singly peaked) initial current sheet is sufficiently wide.