The mechanisms behind the large photocarrier diffusion length and the outstanding photovoltaic performances of lead halide perovskites (LHP) are still debated. Hot carrier cooling and the formation of polarons are central to the charge separation process. Ultrabroadband terahertz spectroscopy with a time resolution <200 fs is applied to probe the early carrier mobility dynamics in LHP samples composed of various anions and cations. The increase of the mobility observed in the subpicosecond time scale evidences the competition between carrier cooling and a dynamic screening process associated with polaron formation. The dependence of the mobility dynamics on the initial photocarrier density and excess excitation energy shows that carrier cooling is not only subjected to a phonon bottleneck but also competes with the dynamic screening effect. A kinetic model is established that successfully unifies the two previously proposed mechanisms. Simulations based on this evidence a chemical composition dependence of the dynamic screening lifetime.