We present a study of the Oosterhoff (Oo) dichotomy in the Galactic bulge using 8141 fundamental-mode RR Lyrae stars. We used public photometric data from the Optical Gravitational Lensing Experiment and the Vista Variables in the Via Lactea survey. We carefully selected fundamental-mode stars without modulation and without association with any globular cluster located towards the Galactic bulge. Subsequently, we identified and separated the Oosterhoff groups I and II on the basis of their period-amplitude distribution and using a relation fitted to the Oosterhoff I locus. Both Oosterhoff groups were then compared to observations of two bulge globular clusters and with models of stellar pulsation and evolution. We found that some of the variables classified as Oo II belong to a third Oo group. The Oosterhoff II variables are more metal-poor on average, more massive, and cooler than their Oosterhoff I counterparts. The analysis of their spatial distribution shows a systematic difference between foreground, central, and background regions in the occurrence of the Oosterhoff II group. The difference between the Oo I and II groups is also seen in their distance distributions with respect to the Galactic bar, but neither group is associated with the bar.