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Source camera model identification (CMI) and image manipulation detection are of paramount importance in image forensics. In this paper, we propose an L2-constrained Remnant Convolutional Neural Network (L2-constrained RemNet) for performing these two crucial tasks. The proposed network architecture consists of a dynamic preprocessor block and a classification block. An L2 loss is applied to the output of the preprocessor block, and categorical crossentropy loss is calculated based on the output of the classification block. The whole network is trained in an end-to-end manner by minimizing the total loss, which is a combination of the L2 loss and the categorical crossentropy loss. Aided by the L2 loss, the data-adaptive preprocessor learns to suppress the unnecessary image contents and assists the classification block in extracting robust image forensics features. We train and test the network on the Dresden database and achieve an overall accuracy of 98.15%, where all the test images are from devices and scenes not used during training to replicate practical applications. The network also outperforms other state-of-the-art CNNs even when the images are manipulated. Furthermore, we attain an overall accuracy of 99.68% in image manipulation detection, which implies that it can be used as a general-purpose network for image forensic tasks.