The main reason, I believe, that there weren't any major conflicts similar to that of Europe of the Middle Ages, is because most Chinese religions never exclusively forbid the practicing other religions. Buddhism in particular, since it had so much influence and was so popular, fits nicely with other religions, in fact. The main goal of Buddhism, roughly, is to escape the suffering of the physical world and achieve enlightenment/nirvana. What happens in the real world, besides trying to achieve enlightenment, is unimportant (more or less, it depends on the sect). So religions such as Daoism, Confucianism, and ancestor worship can exist at the same time with this kind of thinking. Now, how these religions exist together, my most concise and easiest answer would be because they also aren't exclusive and emphasize different aspects of Chinese life and culture. I chose Buddhism to explain though because it was the one more people are familiar with and was the easiest to explain.

As for religious conflicts within China specifically, I'm not too familiar. Shinto is actually a Japanese religion, and it's conflict with Buddhism is mixed in with the Meiji Restoration and state nationalism, which is quite interesting. But back to China, I know there are some instances where probably a few emperors promoted and sponsored Buddhism, but I don't think there were major conflicts? The leader of the Taiping Rebellion in 1864 believed that he was the brother of Jesus, though I don't believe that was the major rallying point for the rebellion.

If you have time for some spare reading, a translation of Journey to the West is an interesting lens to look at how Chinese religions work together. And it's a fun read, also there are several movies and probably video games. The main character, the monkey king, attains Daoist immortality but is still a mischievous monkey, so he needs to be straightened out with Buddhism and so begins a journey with a Buddhist monk to obtain holy scriptures from India.