"X's Y" and "Y of X" have basically the same meaning, they just sound different in different sentences, as others have pointed out in their discussions of the proper uses of the genitive. I'd like to discuss the other form you're using.
The form "X Y", in your example "bird exposure", is NOT forming a possessive. What this does is it creates a compound noun, with the more essential part of the noun coming last (head-final), and the other nouns modifying the final noun.
For example, consider "passenger train" and "freight train". If you were to describe either simply, you would say that it's a train. What's the difference, then? One is a train that carries passengers, the other is a train that carries freight.
Contrast this to "passenger's train". This is still a train, but the train itself has no description except that this passenger "possesses" it in the various ways that the genitive might usually imply. Who is the passenger, though? You might make a construction like this if you were a taxi driver, and someone paid you to take them to the train station.
There was construction on the road, so we didn't make it in time for my passenger's train.
Relating back to your question, the construction "bird exposure" describes a type of exposure. Specifically, an exposure to birds or relating to birds. This kind of sounds like you're describing some adverse effect one might get from being near birds. Taking advantage of this construction to form a relevant sentence, you could try something like:
The pollutant exposure has caused the birds...
Similarly, "feather's formation" would refer to the formation of a feather, while "feather formation" would be a formation made out of feathers, or something similar.