Someone has f**ked up your cooling system electronics bro! Probably a bakapandithaya who doesn't know the cars' design!
Civics have a dual fan system designed specifically for the AC system and Radiator which needs no such alteration. this system is 100% capable of keeping the AC condenser and the engine at the required temp.
Don't run with these makabass setups which will ruin the car in the long run.
This is how the OEM system works.
If engine needs cooling both fans work at their max speed supplying radiator with a cool air flow and both fans shuts off when thermostat engages. (This operation is not related to AC, both fans should run at max speed to quickly cool down the engine)
If the AC kicks-in the passenger side fan works in variable speeds to cool the AC condenser. But if engine needs cooling at the same time both fans kicks in at max speed irrespective of the AC.
This is critical for cooling system. Never alter fan setup, coolant temp sensors or thermostat design.
Anyway follow these steps to diagnose the problem.
1. First scan the car and check for any errors. A bad temp sensor(there are 2, one in the lower radiator area and one up on the engine) or relay/fuse will definitely throw a code.
2. Then check all the wires.
3. Then check if the fan is actually bad by testing it with a ohm meter.
Always keep an eye on the temp gauge as a habit when driving any car.
Even if you replace the whole cooling system it would still be cheaper than a warped head and a blown head gasket!