How do planes fly so high in a thin atmosphere if they need lots of air to generate lift?
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The short answer is speed. The long answer is much more complicated.
Lift is calculated as L=½ρv²S(cl), thus, double the speed 4 times the lift. At high speeds, usually Mach 0.3 and higher, we have to correct the lift coefficient because the air actually compresses.
This new lift coefficient is (cl)=(cl₀)/√(1-M²). This correction works up to M≈0.7, where most aircraft start to go transonic. Because 1/x gets pretty big pretty quick for fractional x, we really just have to worry about maintaining the speed necessary for being in the compressible flow regime.
If we go any faster, the corrections for transonic compressibility require lots of calculus, linear algebra, and looking things up in a database. Supersonic becomes easy again because we can use theories that let us just look things up on a table.
This is where turbine engines really shine. The compressor stage of a turbine engine is incredibly effective at compressing air to produce a combustible mixture (which in turn can be exhausted to produce thrust). These engines also produce an incredible amount of thrust for their size. But because they continuously burn fuel, they burn a lot of it.
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