Assessment of lung disease severity is a crucial step in the clinical workflow for patients with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), the cause for the coronavirus disease 2019 (COVID-19) pandemic. A routine procedure for performing such an assessment involves analyzing chest x-rays (CXRs), with two key metrics being the extent of lung involvement and the degree of opacity. In this study, we introduce COVIDNet-S, a pair of deep convolutional neural networks based on the COVID-Net architecture for performing automatic geographic extent grading and opacity extent grading. We further introduce COVIDx-S, a benchmark dataset consisting of 396 CXRs from SARS-CoV-2 positive patient cases around the world, graded by two board-certified expert chest radiologists (with 20+ years of experience) and a 2nd-year radiology resident. To the best of our knowledge, this is the largest study as well as dataset of its kind for SARS-CoV-2 severity grading. Furthermore, this is the first study of its kind to make both models and dataset open access for the research community. Experimental results using 100-trial stratified Monte Carlo cross-validation (split between geographic and opacity extent) showed that the COVIDNet-S networks achieved R^2 of 0.664 +/- 0.001 and 0.635 +/- 0.002 between predicted scores and radiologist scores for geographic extent and opacity extent, respectively, with the best performing COVIDNet-S networks achieving R^2 of 0.739 and 0.741 for geographic extent and opacity extent, respectively. These promising results illustrate the potential of leveraging deep convolutional neural networks for computer-aided assessment of SARS-CoV-2 lung disease severity.