The thing is before 1.3.0 or I think 1.4.0 handling on controller was better, after that something changed... But you're not alone on that. It's quite unresponsive on controller; movement feels sluggish. But, as Codemasters said (if I recall correctly), it was made like that to simulate the handling realistically, like a wheel (so it actually takes times to turn), although I think this is actually not the best solution. Game is good, but only if you race on a wheel.
Also, yeah, input lag does exist. Sadly even Codemasters are not sure why is that.