Birds regularly experience up 10-14 G according to this website. After looking at how birds and insects fly, scientists came up with the conclusion as to how birds cope with this is because:
Animals will always have some advantages over machines, such as the ability to use their nervous systems to sense subtleties about the environment around them and alter their flight accordingly.
further up near the top, it mentions comparisons between birds and high performance military aircraft, such as this:
A Blackbird jet flying nearly 2,000 mph covers 32 of its own body lengths per second. But a common pigeon covers 75 of its body lengths a second.
The roll rate of the aerobatic A-4 Skyhawk plane is about 720 degrees per second. The roll rate of a barn swallow is more than 5,000 degrees per second.
Some military aircraft can withstand gravitational forces of 8-10 G (Earth's gravity is equal to 1 G). Many birds routinely experience G-forces greater than 10 G and up to 14 G.
According to another website the reason humans can't cope with high g's as well is because:
Every bit of our muscular-skeletal system is naturally crafted to deal with Earth’s gravity. Were you to travel to a planet that had a more significant gravity, a more massive planet with 5 G’s for example, you would either be unable to lift yourself off the ground, our the new weight-force would be so much that the air would be forced out of your lungs and your eyes would explode (something gruesome for sure).