Classical model reduction techniques project the governing equations onto a linear subspace of the original state space. More recent data-driven techniques use neural networks to enable nonlinear projections. Whilst those often enable stronger compression, they may have redundant parameters and lead to suboptimal latent dimensionality. To overcome these, we propose a multistep algorithm that induces sparsity in the encoder-decoder networks for effective reduction in the number of parameters and additional compression of the latent space. This algorithm starts with sparsely initialized a network and training it using linearized Bregman iterations. These iterations have been very successful in computer vision and compressed sensing tasks, but have not yet been used for reduced-order modelling. After the training, we further compress the latent space dimensionality by using a form of proper orthogonal decomposition. Last, we use a bias propagation technique to change the induced sparsity into an effective reduction of parameters. We apply this algorithm to three representative PDE models: 1D diffusion, 1D advection, and 2D reaction-diffusion. Compared to conventional training methods like Adam, the proposed method achieves similar accuracy with 30% less parameters and a significantly smaller latent space.