This paper reports the optimization of the harvested power and bandwidth of bistable piezoelectric energy harvesters (PEH) by mean of a nonlinear energy extraction circuit: the Parallel Synchronized Switch Harvesting on Inductor (PSSHI). The combination of nonlinear PEH with nonlinear circuits have been explored a few times in the literature. For instance, Huguet et al. analyzed numerically and experimentally the mutual influences of Synchronous Electric Charge Extraction (SECE) interfaces with bistable PEH. In the present study, we propose to derive simple compact models that allow to predict and understand the dynamics of bistable PEH associated with PSSHI circuit. Based on a truncated harmonic balance, we prove that the harvested power as well as the bandwidth of the system largely depend on the value of the load resistance connected to the PSSHI circuit. In order to verify these models, a prototype of strongly coupled bistable PEH has been designed. The PEH is made of an inertial mass, two steel buckled beams, and an amplified piezoelectric actuator (APA) built around a stack of lead titano-zirconate piezoelectric ceramic (PZT) installed in an elliptical metallic shell. In order to perform the experimental measurements, a fully automated testbench has been setup. The electrodes of the bistable PEH have been connected to the PSSHI interface. Depending on the value of the load resistance, the PSSHI circuit induces a damping on the bistable PEH, which has an impact on both the bandwidth and the harvested power. Due to the PSSHI properties, the electrically induced damping can reach much larger values than with other circuits. As a consequence of the latter, the PSSHI circuit allows to tune the bandwidth as well as the harvested power of the system on a large frequency band (76.60 Hz instead of 64.84 Hz with fine tuning of a simple resistance) which makes this circuit particularly interesting to enhance bistable PEH performance.