Spatiotemporal modeling of real-world data poses a challenging problem due to inherent high-dimensionality, measurement noise, and expensive data collection procedures. We present \textbf{S}parse \textbf{I}dentification of \textbf{N}onlinear \textbf{Dy}namics with \textbf{SH}allow \textbf{RE}current \textbf{D}ecoder networks (SINDy-SHRED), a method to jointly solve the sensing and model identification problems with simple implementation, efficient computation, and robust performance. SINDy-SHRED uses Gated Recurrent Units (GRUs) to model the temporal sequence of sensor measurements along with a shallow decoder network to reconstruct the full spatiotemporal field from the latent state space using only a few available sensors. Our proposed algorithm introduces a SINDy-based regularization; beginning with an arbitrary latent state space, the dynamics of the latent space progressively converges to a SINDy-class functional, provided the projection remains within the set. Thus a dynamical system model is enforced in the latent space of the temporal sequence model. We conduct a systematic experimental study including synthetic PDE data, real-world sensor measurements for sea surface temperature, and direct video data. With no explicit encoder, SINDy-SHRED enables efficient training with minimal hyperparameter tuning and laptop-level computing; further, it demonstrates robust generalization in a variety of applications with minimal to no hyperparameter adjustments. Finally, the interpretable SINDy model of latent state dynamics enables accurate long-term video predictions, achieving state-of-the-art performance and outperforming all baseline methods considered, including Convolutional LSTM, PredRNN, ResNet, and SimVP.