That would seem to have been one of our disadvantages that we one in spite of, not because of. I've seen former German pilots talking in interviews about using a dogfighting technique of first climbing above an enemy plane and then diving back at it because they knew that Allied planes didn't have the engine power to climb like they could. And the Thach Weave that someone else mentioned before was explicitly designed to overcome enemy planes' superior climb rates and turning rates.
Eh? German fighters (sometimes) had an advantage in rate of climb because they were pitting lightweight and lightly loaded point defense interceptors optimized for climb against much more armored Allied escort fighters, carrying more ammunition for extended combat, and heavily laden with fuel so they could fly so far and still make it home.
It had nothing to do with engines, and indeed the Allies were consistently ahead in both piston and jet engine techology during the war. The P&W R-2800, for instance, was both more reliable than, and had a higher power to weight ratio than the BMW 801 or Jumo 213. The problem is only worsened by the much higher quality fuel consistently available to the Allies. The Germans tried to make up for this with heavily adulterated fuel, which reduced range and engine lifespan, and temporary cheats like nitrous injection which further reduce reliability and still barely broke even by making their aircraft much more lightly loaded than the Allies were comfortable with.
As for the Thatch Weave, the Japanese gained climb and turning rates by stripping absolutely everything from Zeros and Oscars, including even radios, something they resorted to because they were unable to produce engines of sufficient power or fuel better than 87 octane. An advantage bought at great cost and lost once the Americans came out with the Hellcat and Corsair, each packing over twice the horsepower giving them equivalent or superior performance while still carrying heavy armor and armament.