The censored measurements are unavoidably encountered in practical scenarios due to sudden changes of circumstances and physical constraints, and the traditional parameter and state estimation methods may result in obvious estimation bias and serious performance degradation when the collected measurements contain censored observations. This paper concerns the synchronous parameter and state estimation for the bilinear state-space systems with censored measurements. By compensating the estimation bias from the censored measurements and constructing a novel criterion function, a censored regression-based gradient method is presented for synchronously estimating the system parameters and states. Moreover, a censored regression-based multi-innovation gradient method is derived to enhance the estimation performance. Theoretical analysis reveals that the convergence of the parameter estimates can be guaranteed under the persistent excitation condition. The simulation examples exhibit that the proposed method performs significantly better than the conventional auxiliary model-based least mean square method.