Infectious diseases originating from animals (zoonotic diseases) have emerged following deforestation from agriculture. Agriculture can reduce its land use through intensification, i.e., improving resource use efficiency. However, intensive management often confines animals and their wastes, which also fosters disease emergence. Therefore, rising demand for animal-sourced foods creates a "trap" of zoonotic disease risks: extensive land use on one hand or intensive animal management on the other. Not all intensification poses disease risks; some methods avoid confinement and improve animal health. However, these "win-win" improvements alone cannot satisfy rising meat demand, particularly for chicken and pork. Intensive poultry and pig production entails greater antibiotic use, confinement, and animal populations than beef production. Shifting from beef to chicken consumption mitigates climate emissions, but this common strategy neglects zoonotic disease risks. Preventing zoonotic diseases requires international coordination to reduce the high demand for animal-sourced foods, improve forest conservation governance, and selectively intensify the lowest-producing ruminant animal systems without confinement.