SUBJECT IN THE SUN, BACKGROUND IN THE SHADE
You need to be aware that cameras don’t “think” the way we do. A camera doesn’t know this is a bird, or even that it is the main subject of the photo. When it comes to measuring exposure, the camera just looks at all the light and shadow in the view and makes its best guess. In a situation like this, the camera can be fooled by the amount of dark shadow in the shot. It may want to let more light in to avoid underexposing the shadows...and end up overexposing the bird.
This is why it is essential to have control of your exposure. With the camera in charge, if it wants to give you an overexposed bird, that’s exactly what you will get. Here you need to take a photos that is darker than what the camera wants. My approach – and in my opinion the best approach – is to shoot in Manual mode. Work out what the camera thinks is the correct exposure using the light-meter, and then set the shutter speed a couple of stops faster. Your background may become very dark as a result, but that’s OK, it only serves to throw a virtual spotlight on the subject.