We characterize the two-dimensionalization process in the turbulent flow produced by an impeller rotating at a rate ω in a fluid rotating at a rate Ω around the same axis for Rossby number Ro=ω /Ω down to 10-2. The flow can be described as the superposition of a large-scale vertically invariant global rotation and small-scale shear layers detached from the impeller blades. As Ro decreases, the large-scale flow is subjected to azimuthal modulations. In this regime, the shear layers can be described in terms of wakes of inertial waves traveling with the blades, originating from the velocity difference between the nonaxisymmetric large-scale flow and the blade rotation. The wakes are well defined and stable at low Rossby number, but they become disordered at Ro of order of 1. This experiment provides insight into the route towards pure two-dimensionalization induced by a background rotation for flows driven by a nonaxisymmetric rotating forcing.