Point the first: not everyone is good at every game, even games in the same field. For example, while I am pretty dang good at L4D/L4D2, I am the terribles at TF2. Two multiplayer FPS games, even made by the same people; wildly different performance. Maybe this just isn't your game.
Point the second: expecting not-easy difficulty levels to allow you the same slack in gameplay as easy did is always, always, always a mistake. You should not grow accustomed to getting shot just because you can take it on easy; the only reason they make you capable of sustaining that kind of damage without immediately losing on easy is so that you can learn how to start avoiding that kind of damage without the frustration of constant You Lose screens, not so you learn that it's okay to get shot.
As for whether Arkham City is actually an easy game or not, couldn't tell you, I haven't played it yet (installed, but not played). That said, given its wealth of similarities to Arkham Asylum, which I must say I did find to be a very easy game, yes, it's probably on the low end of the "overall" difficulty spectrum as games go. In AA the only thing I found challenging at all were the challenge modes, and even in those it was only challenging to do the combos required to rack up a high score, not to just "win" a given challenge (exception: the one with the insta-lose electric floor timer; that arena was bullshit).
That said, AA's battle system plays to one of my strengths, that being counter-attacks and chaining combat. I really enjoy counter systems, so the Batman games are basically designed for me, so of course I'd find them easy to play. That's just me, not necessarily indicative of their "objective" difficulty.