We present new observations of a controlled transition to turbulence in a fundamental but little-studied regime: circular Couette flow with only the outer cylinder rotating. Our apparatus consists of an outer cylinder of fixed radius and three inner cylinders having different radii that are used interchangeably to study the effect of flow curvature. With the smallest inner cylinder the end-cap configuration (vertical boundary conditions) may also be varied. The turbulent transition is found to be sensitive to both gap width and end-cap configuration, with wider gaps transitioning at higher rotation rates. All configurations are observed to transition with hysteresis and intermittency. A laser Doppler velocimetry (LDV)-based study of the azimuthal velocity profile as a function of gap width and rotation rate reveals that turbulence, once initiated, is confined to regions of significant shear. For wider gap widths, the radial location of these shear layers is determined by the chosen end-cap configuration. This, in turn, affects the transition Reynolds number, which we posit to be radially dependent. The narrow-gap case in particular features spiral turbulence, whose properties are found to be similar to observations of the phenomenon in related shear flows. The velocity profile in this case is correlated with overlapping boundary layers, suggesting a coupling mechanism for the origin of laminar-turbulent banding phenomena.