In this article, we present Bayesian estimation of a shifted exponential distribution assuming progressive type-II censoring with random removals. Bayes estimators and their respective posterior risks are derived under the squared error loss function and the precautionary loss function. Moreover, the credible intervals and expected experiment time along with the Bayesian predictive intervals for one- and two-sample cases are also discussed. A comparison of the expected experiment time under complete and type-II progressive censoring is also a part of this study. Finally, a real-life example is provided to illustrate the proposed methodology.