We apply density functional theory for superconductors (SCDFT) to doped tungsten oxide in three forms: electrostatically doped WO₃, perovskite WO_3−xF_x, and hexagonal Cs_xWO₃. We achieve a consistent picture in which the experimental superconducting transition temperature T_c is reproduced, and superconductivity is understood as a weak-coupling state sustained by soft vibrational modes of the WO₆ octahedra. SCDFT simulations of Cs_xWO₃ allow us to explain the anomalous T_c behavior observed in most tungsten bronzes, where T_c decreases with increasing carrier density. Here, the opening of structural channels to host Cs atoms induces a softening of strongly coupled W-O modes. By increasing the Cs content, these modes are screened and T_c is strongly reduced.