Mediation analysis assesses the effect of study exposures on an outcome both through and around specific mediators. While mediation analysis involving multiple mediators has been addressed in recent literature, the case of multiple exposures has received little attention. With the presence of multiple exposures, we consider regularizations that allow simultaneous effect selection and estimation, while stabilizing model fit and accounting for model selection uncertainty. In the framework of linear structural-equation models, we analytically show that a two-stage approach regularizing regression coefficients does not guarantee a unimodal posterior distribution and that a product-of-coefficient approach regularizing direct and indirect effects tends to penalize excessively. We propose a regularized difference-of-coefficient approach that bypasses these limitations. Using the connection between regularizations and Bayesian hierarchical models with Laplace prior, we develop an efficient Markov chain Monte Carlo algorithm for posterior estimation and inference. Through simulations, we show that the proposed approach has better empirical performances compared to some alternatives. The methodology is illustrated using data from epidemiological studies in human reproduction.