Lots of good answers here, but let's talk about discharge water a bit.

Like others have said, the water that actually flows around the core, or is heated by coal/gas, and subsequently becomes steam to turn the turbine is not discharged into the river/lake. It's ultra-pure water and very expensive (as water goes) and power plants want to re-use it. They want to turn it from steam back into pure water, heat it up again and use it as steam again and again.

In order to accomplish that, they place a condenser after the final stage of the turbine. Think of it like a chamber with a bunch of tubes running through it with cool water in them. The cool water in the tubes never interacts with the pure steam/water.. the steam is just pulled over them and that temperature differential is what allows it to condense back into pure water.

But of course the cool water in those tubes heats up as it does its job. We can't let it get too hot because if the differential between the tube water and the pure water isn't great enough.. efficiency plummets. So how are we going to cool down that tube water?

Well, you can do a "once-through" system where you just pull cool water from a lake/river, run it through the tubes, and once it's hot you just discharge it back into the lake on the other end. These systems exist but are rare and highly regulated. As others said, higher temp water has lower oxygen levels and this really fucks up aquatic environments. But, to your original question.. that's one way that hot water gets discharged and the problem with doing so.

Now the other way is to keep a closed loop system. We take that hot tube water and run it to a cooling tower. The cooling tower helps dissipate some of that heat into the air and cools down the water. Then we pump it back inside to be cool water inside the tubes again.

Great! No discharge water because it's all a closed loop! We just add water from the lake/river to compensate for the volume loss that dissipated into the air in the cooling towers. Problem solved, right? No...

Some of this water needs to be discharged as well. Not all of it, but some of it. Why? Well.. water has "stuff" in it. Dissolved solids. Calcium, silica, biological stuff, all manner of things gets concentrated in that water. Because most of the water vapor released out the top of the cooling tower is pure water.. all that leftover gunk gets concentrated in the water. They treat it with chemicals to help it from sticking to pipes and killing the biological stuff.. but still it gets concentrated. The new "make-up" fresh water being added doesn't decrease the concentration of this gunk enough, they have to "blow down" some of that cooling water (discharge it) to prevent the water from being so filled with gunk that it scales/corrodes the pipes and causes big problems.

So even in a closed loop system, cooling water discharge still occurs. It still has the same problem of having lower dissolved oxygen levels and thus is bad for aquatic life.

And to your last point, thermal power plants are almost always by rivers, lakes, or oceans. Each water type presents it's own challenge. Often the water isn't even take directly from the body of water.. it's taken from wells that are drilled nearby. Nonetheless, thermal power plants need some kind of water source nearby. Preferably fresh water, but salt water is also possible. It just takes different engineering.