To me, the A/C is working normally.

What car?
What setting did you set the A/C thermostat to?

On a hot day, it's normal for the A/C to work harder to maintain the temperature inside the cabin. But no matter what, the A/C should not cause the engine to overheat. Something else is then inherently wrong with the engine's cooling system. Check radiator fans, coolant level and condition, engine thermostat valve condition, radiator, etc.