We investigate the intraband nonlinear dynamics in doped bilayer graphene in the presence of strong, linearly polarized, in-plane terahertz fields. We perform degenerate pump-probe experiments with 3.4-THz fields on doped bilayer graphene at low temperature ($12\phantom{\rule{0.16em}{0ex}}\mathrm{K}$) and find that when the pump is copolarized with the probe beam, the differential pump-probe signal is almost double that found in the cross-polarized case. We show that the origin of this pump-induced anisotropy is the difference in the average electron effective mass in the probe direction when carriers are displaced in $k$ space by the pump either parallel or perpendicular to the direction of the probe polarization. We model the system using both a simple semiclassical model and a Boltzmann equation simulation of the electron dynamics with phenomenological scattering and find good qualitative agreement with experimental results.