This article proposes an online scheme for state estimation of a generic class of nonlinear dynamical systems in the presence of abnormal measurement data from sensors. We thoroughly illustrate why the performance of standard recursive Bayesian inference degrades in the presence of any measurement distortion. After demonstrating how different abnormalities can be accommodated explicitly inside a generic state-space model, we propose a robust mechanism to perform recursive Bayesian inference on the presented model to not only detect but also mitigate the effect of corrupted measurements in the final state estimates. Using simulations and experimentation, we demonstrate the success of the proposed framework in reducing the impact of different types of distortions in measurements. The ability to tackle different kinds of measurement abnormalities during online inference sets the proposed method apart from the existing techniques.