Adding time delay to nonlinear systems can significantly improve their performance, making it possible to construct image-encryption algorithms with higher security. In this paper, we propose a time-delayed nonlinear combinatorial hyperchaotic map (TD-NCHM) with a wide hyperchaotic interval. Based on TD-NCHM, we develop a fast and secure image-encryption algorithm that includes a plaintext-sensitive key-generation method and a simultaneous row-column shuffling-diffusion encryption process. Plenty of experiments and simulations demonstrate the superiority of the algorithm in terms of efficiency, security, and practical value in secure communications.