Recurrent stochastic configuration networks (RSCNs) have shown great potential in modelling nonlinear dynamic systems with uncertainties. This paper presents an RSCN with hybrid regularization to enhance both the learning capacity and generalization performance of the network. Given a set of temporal data, the well-known least absolute shrinkage and selection operator (LASSO) is employed to identify the significant order variables. Subsequently, an improved RSCN with L2 regularization is introduced to approximate the residuals between the output of the target plant and the LASSO model. The output weights are updated in real-time through a projection algorithm, facilitating a rapid response to dynamic changes within the system. A theoretical analysis of the universal approximation property is provided, contributing to the understanding of the network's effectiveness in representing various complex nonlinear functions. Experimental results from a nonlinear system identification problem and two industrial predictive tasks demonstrate that the proposed method outperforms other models across all testing datasets.