It actually does, that’s called the correolis effect, and it effects every flying object. Snipers & artillerymen actually have to do the math for that on their shots or they’ll miss.
But, when you took off, you (and the air around you) were already moving in the same speed & direction as the ground, so momentum is still carrying you in that direction at that speed. The earth does move under you, but it is not moving very fast relative to your initial velocity, so you normally wouldn’t notice it unless you’re at a really high altitude or you’re taking very precise measurements.
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