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We present new experimental results from the University of Maryland Three Meter Geodynamo experiment. We drive a fully turbulent flow in water and also in sodium at magnetic Reynolds number Rm= ΔΩ(ro−ri)2/η, up to 715 (about half design maximum) in a spherical Couette apparatus geometrically similar to Earth's core. We have not yet observed a self‐generating dynamo, but we study MHD effects with an externally applied axisymmetric magnetic field. We survey a broad range of Rossby number −68 < Ro= ΔΩ/Ωo< 65 in both purely hydrodynamic water experiments and sodium experiments with weak, nearly passive applied field. We characterize angular momentum transport and substantial generation of internal toroidal magnetic field (the Ωeffect) as a function of Roand find a rich dependence of both angular momentum transport and Ωeffect on Ro. Internal azimuthal field generation peaks at Ro= 6 with a gain as high as 9 with weak applied field. At this Rossby number, we also perform experiments with significant Lorentz forces by increasing the applied magnetic field. We observe a reduction of the Ωeffect, a large increase in angular momentum transport, and the onset of new dynamical states. The state we reach at maximum applied field shows substantial magnetic field gain in the axial dipole moment, enhancing the applied dipole moment. This intermittent dipole enhancement must come from nonaxisymmetric flow and seems to be a geodynamo‐style feedback involving differential rotation and large‐scale drifting waves. Many states in turbulent spherical Couette flow depending on Rossby numberTransitions affect angular momentum transport and internal field generationDynamo‐like enhancement of applied external dipole by nonaxisymmetric waves