You're confusing reproduction ratio with zoom ratio. When people say a 150-600mm is a 4x zoom, they mean that the ratio of the longest focal length (600mm) to the shortest (150mm) is 4:1, or 4 times. It has nothing to do with how large of an image of an object the lens will make on a sensor. That's reproduction ratio. For example, a repro ratio of 1:2 means the lens will project an image of an object on a sensor that is exactly half as wide (or half as tall) as the actual object. Note that this is a maximum. Any lens can of course also project smaller images if you just put more distance between the object and the camera.
The maximum reproduction ratio of the 70-300mm lens you have is 0.22x, or 1:4.5. This repro ratio applies only at 300mm and only at the closest focus distance. So if you zoom in to 300mm and get as close to a bird as you can while still being able to focus on it (1.1 meters for this lens; that's a lot closer than 15 yards), you will get an image of the bird that is almost 1/4 actual size. That might be enough to fill the DX frame with the bird head. If it's a big enough bird.