In this paper, we study the large-sample properties of the posterior-based inference in the curved exponential family under increasing dimensions. The curved structure arises from the imposition of various restrictions on the model, such as moment restrictions, and plays a fundamental role in econometrics and others branches of data analysis. We establish conditions under which the posterior distribution is approximately normal, which in turn implies various good properties of estimation and inference procedures based on the posterior. In the process, we also revisit and improve upon previous results for the exponential family under increasing dimensions by making use of concentration of measure. We also discuss a variety of applications to high-dimensional versions of classical econometric models, including the multinomial model with moment restrictions, seemingly unrelated regression equations, and single structural equation models. In our analysis, both the parameter dimensions and the number of moments are increasing with the sample size.