It isn't a you problem. It's a geometry problem, and it's built into the window:
Outward-opening windows put the hardware on the wrong side.
Casement, awning, hopper and tilt-and-turn sashes swing into the outside space — so the screen has to mount inside, in front of the crank. Roughly a third of the windows sold in the US work this way. Every one of them makes you choose between the screen and the handle.
The bugs get in during the thirty seconds it's off.
Not while the window is open. While the screen is leaning against the wall and your hands are full. That's the window mosquitoes actually use.
The fix that exists is sold to screen shops, not to you.
It's called a wicket, it costs about nine dollars, and it only fits while a screen is being built — spline out, mesh re-stretched, spline back in. Which is why the quote you got was for a whole new screen.
Every removal is a chance to bend the frame.
Thin aluminium, popped spline, a torn corner. Four summers of daily handling and you're buying the replacement screen anyway.
It isn't that you're opening your windows wrong.
It's that your screen doesn't have a door in it.