As one of the representative models in the field of image generation, generative adversarial networks (GANs) face a significant challenge: how to make the best trade-off between the quality of generated images and training stability. The U-Net based GAN (U-Net GAN), a recently developed approach, can generate high-quality synthetic images by using a U-Net architecture for the discriminator. However, this model may suffer from severe mode collapse. In this study, a stable U-Net GAN (SUGAN) is proposed to mainly solve this problem. First, a gradient normalization module is introduced to the discriminator of U-Net GAN. This module effectively reduces gradient magnitudes, thereby greatly alleviating the problems of gradient instability and overfitting. As a result, the training stability of the GAN model is improved. Additionally, in order to solve the problem of blurred edges of the generated images, a modified residual network is used in the generator. This modification enhances its ability to capture image details, leading to higher-definition generated images. Extensive experiments conducted on several datasets show that the proposed SUGAN significantly improves over the Inception Score (IS) and Fréchet Inception Distance (FID) metrics compared with several state-of-the-art and classic GANs. The training process of our SUGAN is stable, and the quality and diversity of the generated samples are higher.