Here are some possible reasons why your car's air conditioning system might only work for a few minutes after the car is turned on:

Low Refrigerant Levels: Insufficient refrigerant in the AC system can cause it to blow cold air initially but lose effectiveness after a few minutes as the refrigerant pressure drops.

Leak in the System: A leak in the AC system can result in a gradual loss of refrigerant, which can lead to reduced cooling capacity over time.

Faulty Compressor: The compressor is responsible for circulating refrigerant through the system. If malfunctioning, it may only work correctly for a short period before failing to maintain cooling.

Clogged or Blocked Condenser: The condenser removes heat from the refrigerant. If it's clogged with debris or blocked by another component, it can hinder proper cooling performance.

Faulty Condenser Fan: The condenser fan helps dissipate heat from the condenser. If it's not functioning correctly, the condenser may overheat, reducing the efficiency of the AC system.

Faulty Expansion Valve or Orifice Tube: These components regulate refrigerant flow through the system. If they malfunction, it can affect the AC system's cooling capacity.

Electrical Issues: Faulty wiring, relays, or sensors can disrupt the proper functioning of the AC system, leading to intermittent cooling or failure after a few minutes.

Overheating: If the engine or AC components are overheating, it can trigger safety mechanisms that temporarily shut down the AC system to prevent damage.

Moisture or Contamination: Moisture or contaminants in the AC system can interfere with the proper operation of components, leading to reduced cooling efficiency.

Improper Maintenance: Lack of regular maintenance, such as cleaning or replacing air filters, can reduce the efficiency of the AC system over time.

It's essential to have a qualified mechanic inspect and diagnose the issue to determine the exact cause of the problem and perform any necessary repairs or maintenance.