Humans get sick when drinking dirty water, so how do wild animals survive drinking mainly dirty water?

It depends on your definition of "dirty water."

When agriculture was invented, it created large numbers of people in one place. This pollutes water, mostly because humans shit all over the place and then dump it in the river. This happened to a much greater extent when we industrialized, with coal dust and other toxins getting into the water supply. Which we now need to (and do) filter out.

This wasn't a concern for most of Earth's existence, and, in many remote places, still isn't a concern. (I'm thinking Arctic here, there's no significant human or industrial presence.)

It's also a matter of immune system. If you're drinking unfiltered water your entire life, you're going to build up a natural resistance to whatever bacteria may be in that water. This can be seen in modern-day - when many tourists go to somewhere with lower water quality, they'll get stomachaches from the water because they're not used to it. (I had that problem when I spent a summer in Warsaw, frex.)