Generative Adversarial Networks (GANs) significantly advanced image generation but their performance heavily depends on abundant training data. In scenarios with limited data, GANs often struggle with discriminator overfitting and unstable training. Batch Normalization (BN), despite being known for enhancing generalization and training stability, has rarely been used in the discriminator of Data-Efficient GANs. Our work addresses this gap by identifying a critical flaw in BN: the tendency for gradient explosion during the centering and scaling steps. To tackle this issue, we present CHAIN (lipsCHitz continuity constrAIned Normalization), which replaces the conventional centering step with zero-mean regularization and integrates a Lipschitz continuity constraint in the scaling step. CHAIN further enhances GAN training by adaptively interpolating the normalized and unnormalized features, effectively avoiding discriminator overfitting. Our theoretical analysis firmly establishes CHAIN's effectiveness in reducing gradients in latent features and weights, leading to improved stability and generalization in GAN training. Empirical evidence supports our theory. CHAIN achieves state-of-the-art results in data-limited scenarios on CIFAR-10/100 and ImageNet, and in five low-shot and seven high-resolution few-shot image datasets.