The density dependence of the critical in-plane magnetic field Bc at the bifurcation of the resistivity of two-dimensional electron systems with low levels of disorder is determined using the spin-polarization dependence of the electron exchange-correlation hole. Recent numerical simulation results for ground-state energies also permit determination of the magnetic field Bpol(n) needed to saturate the spin polarization. The resulting picture gives a good account of reported experimental results for Bc as a function of electron density in p-type GaAs systems and indicates that the interactions between electrons play a crucial role in the bifurcation phenomenon.