Turns steepen because the outside (top) wing travels faster than the inside (bottom) wing.
I've heard the same thing in a number of places and accepted it as gospel. But I just did the calculation for a PA-28 with a wingspan of 10.66 meters, doing a standard rate turn at 100 knots. I found that the radius of such a turn is 955 meters, so the speed difference between the wingtips is just a hair over 1.1%. Lift is not generated by the wingtips alone but by the whole wing so measured over the whole wing the speed difference, and therefore the lift difference, is more like 0.5%. I can't believe this is really responsible for the steepening of the turn.
Even in a 60 degree, 2g steep turn at 100 knots the radius of the turn is still 125 meters. This is still less than 5% speed difference between the wingtips, and maybe 2.5% speed/lift difference across the whole wing.
It would seem that even a little turbulence would have more impact than these low numbers. Any comments?