The paper examines the problem of cancellation of direct signal, multipath and clutter echoes in passive bistatic radar (PBR). This problem is exacerbated as the transmitted waveform is not under control of the radar designer and the sidelobes of the ambiguity function can mask targets including those displaced in either (or both) range and Doppler from the disturbance. A novel multistage approach is developed for disturbance cancellation and target detection based on projections of the received signal in a subspace orthogonal to both the disturbance and previously detected targets. The resulting algorithm is shown to be effective against typical simulated scenarios with a limited number of stages, and a version with computational savings is also introduced. Finally its effectiveness is demonstrated with the application to real data acquired with an experimental VHF PBR system.