And this might vary from state to state in the 
United States, but here in Montgomery County in 
Maryland, stray and injured animals (if the owner 
is unknown that is) are brought to an animal 
shelter run by either the county or the state 
(I'm not certain which).  Depending upon the 
extent of the animal's injuries, pain, 
adoptability, likelihood to survive the procedure 
and yes, regrettably, the expense, they might 
simply put the animal down and not attempt to 
treat it at all. 

I would never risk any pet's life expecting the 
animal shelter to take care of his or her 
injuries and then 'adopt' the pet back.  The 
chance that they would put the pet down is too 
high, especially if they do not have any more 
room to board new animals.