Recent works in aerial robotics show that the self-organized and cohesive flight of swarms can emerge from the exchange of purely local information between neighboring agents. However, most of the current swarm models are not capable of flight in densely cluttered environments. Predictive models have the potential to incorporate safe collision avoidance capabilities and give the agents the ability to anticipate and synchronize their trajectories in real-time. Here, we propose a distributed predictive swarm model that generates self-organized, safe, and cohesive trajectories by solving an optimization problem in real-time. In simulation, we show that our method is scalable to large numbers of agents and suitable for deployment in different environments, specifically a forest and a funnel-like environment. Furthermore, our results show that the agents are capable of collision-free flight with noisy sensor measurements for a noise level of up to 70% of the magnitude of the agent safety distance. Real-world experiments with a swarm of up to 16 quadrotors in an indoor artificial environment validate our method. Supplementary Materials can be found at https://doi.org/10.5281/zenodo.5245214.