We present a projection-based nonlinear model order reduction (MOR) method for parametrized transport-dominated partial differential equations that exhibit sharp or discontinuous features such as shocks. Given parameter space D, these problems often yield parametric solution manifolds {u(µ)}µ∈D with slowly decaying Kolmogorov N-widths, rendering classical MOR methods based on linear reduced spaces ineffective. On the other hand, nonlinear approximations based on brute-force neural networks with many parameters often require large training datasets, which may render them infeasible in practical engineering settings. In this work, we construct efficient nonlinear reduced-order models in training-data-limited scenarios by exploiting the composite sparsity of the parametric manifolds associated with transport-dominated problems. Specifically, given µ ∈ D, we construct a two-level solution approximation of the form u(µ) ≈ u_{N1,N2} (α1(µ), α2(µ)) = f1_{N1} (α1(µ)) ◦ f2_{N2} (α2(µ)), where f1_{N1} and f2_{N2} lie in N1- and N2-dimensional function spaces with generalized coordinates α1(µ) and α2(µ), respectively. We efficiently construct f1_{N1} and f2_{N2} using a gradient-based optimization algorithm in the offline stage and find the reduced coefficients α1(µ) and α2(µ) using Galerkin projection in the online stage. We demonstrate the effectiveness of our approach for two-dimensional Burgers’ equation and Navier–Stokes equations with parameter-dependent shocks, emphasizing on assessing them in training-data-limited scenarios.