Large convolutional neural networks (CNN) can be difficult to train in the differentially private (DP) regime, since the optimization algorithms require a computationally expensive operation, known as the per-sample gradient clipping. We propose an efficient and scalable implementation of this clipping on convolutional layers, termed as the mixed ghost clipping, that significantly eases the private training in terms of both time and space complexities, without affecting the accuracy. The improvement in efficiency is rigorously studied through the first complexity analysis for the mixed ghost clipping and existing DP training algorithms.Extensive experiments on vision classification tasks, with large ResNet, VGG, and Vision Transformers (ViT), demonstrate that DP training with mixed ghost clipping adds $1\sim 10\%$ memory overhead and $<2\times$ slowdown to the standard non-private training. Specifically, when training VGG19 on CIFAR10, the mixed ghost clipping is $3\times$ faster than state-of-the-art Opacus library with $18\times$ larger maximum batch size. To emphasize the significance of efficient DP training on convolutional layers, we achieve 96.7\% accuracy on CIFAR10 and 83.0\% on CIFAR100 at $\epsilon=1$ using BEiT, while the previous best results are 94.8\% and 67.4\%, respectively. We open-source a privacy engine (\url{https://github.com/woodyx218/private_vision}) that implements DP training of CNN (including convolutional ViT) with a few lines of code.