We seem to know how birds fly. Mostly it is to do with the obvious, which is the controlled flapping of the wings. Lower overall body density helps. This is because some of the bones in birds are hollow.
A study I have found deals with the structure of bones in a range of birds. The results are quite startling. While the bones are hollow, the density of the bone material is higher, and the bone is also less porous. When compared with some mammals, the proportion of a birds skeleton weight to the other parts remains much the same as a mammal.
This tends to show that a bird has no advantage over a mammal when it comes to flying, except for its wings. But birds do have some other features that mammals don't have. These are quite extensive air sacs in various locations in the body. But carrying around a large sac of air would seem to be of little advantage. Remove the sacs, make the bird smaller and lighter, and it should fly even better.
I suggest we are missing something very important. A very strong hollow bone is a pressure vessel.
An air sac, with muscular control, is a pressure vessel. A feather is a pressure vessel.
Another feature of birds is extremely good thermal regulation. They can control their body temperature.

In the previous few pages on this site I have attempted to show that we are mistaken when we believe that clouds in the sky are held up by updrafts. My belief is that as water vapor  condenses into water droplets, it not only gives off a huge amount of latent heat, but its energy (or voltage, or potential) level changes to that of its surroundings,  and gravity then has no immediate effect on it.
The evaporation and condensation cycle in the atmosphere uses latent heat to moderate temperatures around the planet. We use the same principles in our air conditioning systems, by moving a refrigerant, (which could be water), between pressure vessels. 

So, what has this got to do with a birds ability to fly. Let's say a bird can control more than just body temperature. It would also need to simultaneously control both the humidity and the pressure within all of its pressure vessels, the hollow bones, the air sacs, and the feathers.
When the bird is on the ground it would need to keep its body temperature as high as possible, and the pressures within its cavities as high as possible. This would allow the highest possible humidity (water vapor) within the cavities. Water vapor is known to be lighter than dry air, so the higher the humidity level, the lighter the bird will be when liftoff is required.
Let us now have the bird take to the air. It opens and lifts its wings. The wings open out to the top of their reach. As the bird pushes its wings down it creates a lifting force and clears the ground. It then releases pressure from some or all of its cavities. This would have the effect of instantly condensing the water vapor in the cavities. This would release the latent heat, and is why a bird needs such a highly developed thermal regulation system. Simultaneously the air and water in the cavities, in the form of a cloud, would become weightless. Quite possibly the bird could use some of this latent heat to power its muscles directly especially in the initial upward acceleration. Another possibility is that not just the air and water become weightless, but that this effect is a field which may extend into or completely outside the bird. This would mean that some part or all of the bird may have a substantial weight reduction when flying.
I suspect that this thermal / anti gravitational cycle of condensation and evaporation may be occurring instantaneously on each beat of the bird's wings. My reasoning on this is that clouds in the sky tend to stay at the same altitude where they condense. The bird would need to be constantly changing altitude so would need to constantly renew the condensation cycle. Some of the latent heat could be utilized for an extra spurt of muscular activity, and the rest perhaps just shuffled into and out of the tissues.