Compressed sensing (CS) in magnetic resonance imaging (MRI) enables the reconstruction of MR images from highly undersampled k-spaces, and thus substantial reduction of data acquisition time. In this context, edge-preserving and sparsity-promoting regularizers are used to exploit the prior knowledge that MR images are sparse or compressible in a given transform domain and thus to regulate the solution space. In this study, we introduce a new regularization scheme by iterative linearization of the non-convex clipped absolute deviation (SCAD) function in an augmented Lagrangian framework. The performance of the proposed regularization, which turned out to be an iteratively weighted total variation (TV) regularization, was evaluated using 2D phantom simulations and 3D retrospective undersampling of clinical MRI data by different sampling trajectories. It was demonstrated that the proposed regularization technique substantially outperforms conventional TV regularization, especially at reduced sampling rates.