You have the C-wire connected to the R-wire in the air handler? And power is lost to the thermostat if the thermostat tries to run the air conditioner? If I understand this correctly then you have been shorting out the 24VAC supply to the thermostat with the thermostat. While it may be unlikely there's a possibility that the transformer has been damaged by being shorted out. This may require an electrician or HVAC installer to diagnose and repair.
Some HVAC systems will have two separate 24VAC transformers, one for heat and one for air conditioning. This is especially common if the house originally had only heating with air conditioning added later. To make the distinction between the two transformers there's often "Rh" for the heating transformer and "Rc" for the cooling transformer. By running the C-wire to one of the R-wires you may have shorted out one of the transformers while creating a return path for getting power to the thermostat from the other. When the thermostat closes the circuit for the air conditioner it could have shorted both transformers, or something, which then means the voltage to power the thermostat drops to zero and it powers off.
I'm trying to draw the possible circuits in my head given what was described and one of them tells me that it is possible the two transformers were put in series at some point, meaning what should be a 24VAC circuit had 48VAC placed on it.
One important piece of information needed to diagnose this is knowing how many 24VAC transformers there are. There's almost certainly one in the furnace, or whatever is the primary device for moving air through the system if there's cooling only or something for heat other than something that burns fuel. I don't know if there's a standard location for a 24VAC transformer on air conditioners so I can't really tell you where to check. If there's a humidifier, dehumidifier, or some other accessory device in the HVAC there may be a third 24VAC transformer.
If the thermostat is going dead when it tries to run the air conditioner then there could be a short circuit somewhere, maybe an overloaded transformer (too much load and the voltage drops), or something else bad that could damage something. It is possible nothing has been damaged, at least not yet. Keep guessing on the wires and I expect it won't be long before something expensive breaks.
If you don't want to call in a professional just yet then slow down and connect each device one by one. Perhaps try connecting just the air conditioner first, and make sure that runs as it should. Then add the furnace, check that works. Then add the humidifier or such if there's anything else, then check that.
Most HVAC installers will come out on short notice but they charge extra for that. If you can get the important bits running temporarily (as in keeping the place safe and comfortable for the next day or so) so the pros come out on a scheduled visit, versus an "emergency call", then they charge a lower rate.
I realize that "call a professional" answers are discouraged here but this might be one of those exceptions.