Let's say I'm someone that really hates vaccines. I begin spreading rumors throughout the internet that vaccines cause cancer. My evidence? Most of the people that have cancer had vaccines at one point in the past.

Now how does science disprove this? Through years of study, research and data collection over many decades. Even then, it would be hard to accept or reject a casual relationship between cancer and vaccines, because that is an almost endless set of data. All I would have to do is make up some new casual correlation between vaccines and Y as an attack vector. This is exactly how arguments against climate change are made as well. Science is never really settled, but we can reach a high level of confidence on how something works.