We report on the transport properties of a series of doped 250-nm-thick SrTiO₃ films, grown by molecular beam epitaxy on undoped (001) SrTiO₃ substrates. In particular, we demonstrate anomalies in the superconducting properties. A systematic study of the critical temperatures and fields as a function of the carrier density and type of doping atom reveal deviations from BCS behavior and points towards the role of the specific doping atom. Moreover, the transport properties of the normal state show peaks in the transverse resistance, which indicates an electron nematic phase above the superconducting transition. We discuss the mechanisms of superconductivity of SrTiO₃ thin films.