What are the key topological features of connectivity critically relevant for generating the dynamics underlying efficient cortical function? A candidate feature that has recently emerged is that the connectivity of the mammalian cortex follows an exponential distance rule, which includes a small proportion of long-range high-weight anatomical exceptions to this rule. Whole-brain modeling of large-scale human neuroimaging data in 1,003 participants offers the unique opportunity to create two models, with and without long-range exceptions, and explicitly study their functional consequences. We found that rare long-range exceptions are crucial for significantly improving information processing. Furthermore, modeling in a simplified ring architecture shows that this improvement is greatly enhanced by the turbulent regime found in empirical neuroimaging data. Overall, the results provide strong empirical evidence for the immense functional benefits of long-range exceptions combined with turbulence for information processing.