The difference in wing speeds, as a percentage of the turn radius becomes larger the more steeply you're banked (and the turn becomes smaller.) But this reverses past a certain point, where at very high banks, the wings are nearly on top of each other, and the relative difference shrinks again.
This is only one aerodynamic effect, and there are other minor ones that, as people above have said, there's no pressing need to understand, and can be simply counteracted by use of the controls while maintaining the desired bank, and essentially treated as turbulence or other aberrant minor upsets. For example, is the long-tail effect, where even though the CG of the plane (and the ball, or IRS) is perfectly balanced on the path of the turn circle, the vertical stabilizer is flying outside the circle and sees a relative wind coming from the outside, and therefore produces a yaw (and roll) toward the outside.
Dihedral, by itself, has nothing to do with this though. There is no physical process by which the projected wing area over the ground, effects the amount of lift. Dihedral effect is a response to sideslip and not bank, and will therefore only act on the plane if sideslip occurs.