Dragonflies are pretty much the best flyers among the insects, and they have four wings. Maybe not directly comparable, but still.
Flies in particular are also among the best flyers, and yet, they use only two of their wings for flying, the other two shrank to became halteres, a sensory organ acting like a gyroscope.
Dragonflies may be the best fliers in the insect world thanks to their four independently controlled wings, but flies may be the second best, and they achieved that by losing two of their four wings. Evolution is interesting.
At the current level of O2 in the air I don't believe that flying insects can grow as large as even a hummingbird, much less an eagle.
This got me curious. At a quick search, I found out that there in fact is at least a species of flying insect larger than (the smallest) hummingbird:
https://en.wikipedia.org/wiki/Gauromydas_heros
https://en.wikipedia.org/wiki/Bee_hummingbird
The insect grows up to 7 cm in length whereas the bird can stay less than 6 cm at adulthood. No information about the insect weight though (for a proper comparison).