The usual problem is that the image is properly exposed but the bird is not. The exposure is too low on the bird because the meter is seeing the sky. The camera software then tries to bring up the bird but the underexposure on the bird means there isn't enough data.

The simple answer is exposure compensation. When you have a backlit subject, overexpose by 1 or even 2 stops. Using a narrower area for the exposure sensing helps as well. There are various options available depending on your camera. Sometimes the spot metering is too narrow, but a small area works better than the default wide area. It takes some practice with your setup to find the right settings that work most often. Sometimes the narrowest meter setting comes at a price of focus speed. When a bird is perched you have time to play with things and can do better, but you need to find a configuration that is the one you want when you only get one chance. That is the one for your first shot and should be the one you have the camera in as you walk around. Learn how to set the exposure compensation quickly because not all birds are backlit.

Shooting RAW can help because you can pull up the shadows as much as you want instead of letting the camera decide. You need software that can interpret RAW files and 5x more room on your memory card. I went a long time just shooting JPEG and using compensation, but RAW really works better.