Use the heat to boil water to create steam, run the stream through a steam turbine, and condense the steam back to water. This water is continually recycled throughout the system.
The river water is used to condense the steam back to water and then the water is discharged back to the river - warmer than it came in. If the water coming in from the river is too warm then the condensation rate increases until you get to the point it's out of spec. You can't condense the water fast enough. You need to reduce the plant's output, i.e. reduce the heat.
On the other end the discharge water is always warmer than the intake water. As the intake water warms then the discharge water will also warm - all other things being equal. You'll get to the point the discharge water will raise the water temperature to the point where aquatic life is negatively impacted. There are also laws mandating the maximum temperature for the discharge water.
So, you have to reduce output. That's just how these plants work.
That's the nice thing about natural gas plants - the gas turbines are essentially jet engines - they're fueled directly, no steam or cooling required.
This is something rarely talked about in the solar and wind discussion. People love to point out that the wind and the sun provide intermittent power, while ignoring more and more traditional power plants are curtailing output as water temperatures rise or water levels lower (water intake pipes would be exposed).