How can jet engines be more efficient than propellers?
Why aren't propellers way larger than they typically are?


There are some "propellers" these days with a 400 foot disk diameter, known as windmills.
But, they are doing their best to extract energy from the airstream and here lies the issue.

"Relative wind" is conveniently used to combine the net direction and force of two aerodynamic effects, but let's look at an old question of:  can a sail boat ever sail faster than than wind?  This is very easy to explain by separating the "wind" components, rather than combining them.  Ice boats can go several times faster than the wind by using wind across the beam as "thrust", and using "wind" from the motion of the boat to create lift.

This is exactly how powered aircraft fly, the ice boat is simply doing it on its side (hence the tiller is the "elevator").

Which brings us to propellers, which can be more accurately described as generating lift than "moving air".  But, as the graph shows, net thrust of props decreases as speed increases because disk drag increases with speed.

Turbojets become more efficient at higher speeds than props because they have much less disk drag subtracting from their thrust.  Even though they are less efficient at producing thrust, their net thrust out put is much more linear over speed as "ram effect" (increasing amount of air available to produce more thrust) and drag cancel each other out, while the propeller falls victim to its own disk drag.