Why doesn't the Bessemer process oxidize the iron?
I thought you asked a great question, one that I have wondered about myself.

The answer, as best I could determine, is that the iron does oxidize.  A good description of the Basic Oxygen Steelmaking process, today's descendent of the Bessamer process, says so anyway.  The BOS process uses pure oxygen instead of the original Bessamer processes's air.  It reports that some iron does oxidize.  

The oxidation of carbon to form carbon monoxide is more thermodynamically favorable than the oxidation of iron, so it occurs preferentially.  The page I found (once again the link is here) mentions using on the order of 1800 scf of oxygen per ton of metal.  A ton of steel is about 18 kmol of iron.  1800 scf of oxygen is about 2 kmol of oxygen.  So there is not enough oxygen to oxidize even close to all the iron in the average BOS run.  That fact, combined with the fact that carbon, silicon, and other impurities in the pig iron oxidize preferentially, mean that not much iron is oxidized during BOS or Bessamer.  

But "not much" does not mean zero.  My guess is that iron oxides are more soluble in the slag than they are in molten iron and so are also removed with the slag after BOS.