The key issue is defining "performance."
Thanks to reduced drag at higher altitudes (up to a point) the aircraft can travel faster (true airspeed). Even with reduced power and thrust (less air to burn with fuel, and less air to push for thrust).
And has to travel faster, to generate enough lift in the thinner air.
TAS is not the only measure of performance, however.
A takeoff at a high altitude will take longer - less thrust for acceleration, and more TAS needed, to reach rotation speed measured in IAS.
If you have a "drag race" between two identical planes taking off - one at sea level and one from 6,000 feet (2,000 meters) - the plane at sea level will have much better performance - a shorter takeoff roll in both time and distance, and better climb performance (obstacle/terrain clearance capability) after liftoff. Accelerating at the low end of the speed range (from zero to Vr/V2), the lower drag of thinner air doesn't help performance much.