Two-headed animals are the same as conjoined twins in humans, and would probably be better described as two animals sharing the same body. From what I can tell, there are a few confirmed cases of conjoined birds, but it is rare that they are found and unlikely that they would survive to adulthood, much less fly.
Here is a news report that appears to show a set of conjoined twin barn swallows, although they are only joined by a small amount of tissue, so cannot really be called a “two-headed bird”:
Here is an example of a “two-headed” Northern Cardinal chick, although it’s unclear to me whether it truly has two heads or one malformed head. At any rate it’s unlikely that it survived to adulthood.
In human conjoined twins, such as often each twin controls the limbs on his or her half of the shared body. I suspect that in birds this might make flight impossible. The Hensel twins can coordinate their actions to do complex tasks like typing, walking, riding a bicycle, and driving, but birds’ brains do not have the same level of flexibility as those of humans, and birds cannot use language to plan and coordinate their actions the way humans can.
There are many images of two-headed birds on the Internet, but most are the result of creative Photoshopping or creative taxidermy.