Distributionally robust chance-constrained programs (DR-CCP) over Wasserstein ambiguity sets exhibit attractive out-of-sample performance and admit big-$M$-based mixed-integer programming (MIP) reformulations with conic constraints. However, the resulting formulations often suffer from scalability issues as sample size increases. To address this shortcoming, we derive stronger formulations that scale well with respect to the sample size. Our focus is on ambiguity sets under the so-called left-hand side (LHS) uncertainty, where the uncertain parameters affect the coefficients of the decision variables in the linear inequalities defining the safety sets. The interaction between the uncertain parameters and the variable coefficients in the safety set definition causes challenges in strengthening the original big-$M$ formulations. By exploiting the connection between nominal chance-constrained programs and DR-CCP, we obtain strong formulations with significant enhancements. In particular, through this connection, we derive a linear number of valid inequalities, which can be immediately added to the formulations to obtain improved formulations in the original space of variables. In addition, we suggest a quantile-based strengthening procedure that allows us to reduce the big-$M$ coefficients drastically. Furthermore, based on this procedure, we propose an exponential class of inequalities that can be separated efficiently within a branch-and-cut framework. The quantile-based strengthening procedure can be expensive. Therefore, for the special case of covering and packing type problems, we identify an efficient scheme to carry out this procedure. We demonstrate the computational efficacy of our proposed formulations on two classes of problems, namely stochastic portfolio optimization and resource planning.