Goering bald-faced lied that he was involved until they Nuremberg lawyers produced evidence showing he was lying. At that point, he was trying to convince the tribunal that 'everybody does it', once the 'I had no idea this was happening' and 'I was only following orders' approaches failed.
The camps weren't exactly a well-kept secret. What was happening IN the camps WAS. The Allies didn't believe that the Germans were doing the things they were doing and really didn't have time to investigate. Remember, the camps started back in 1933. But even when the Allies got detailed reports of the Nazi's activities, they rejected them as exaggeration. Though I'm not sure exactly what they could do in the midst of the war that would be different.
As it was, when the allies actually discovered what was going on when they liberated the camps, some guards were killed by the US forces in a rage, which was the subject of a disciplinary review, later.