Thanks for the answers everyone. After thinking through several of the examples, I think I have it now. Please let me know what you think. I'll start with the basics of turning just to make sure I am not making the wrong assumptions. Let's at least assume a constantly-banked level turn:

The aircraft turns due to the horizontal or sideways force induced from banking the aircraft.

The horizontal component of lift changes the direction of the relative wind which is now to the left or right of the nose of the aircraft. It could be said that the direction of the relative wind is directly opposite the flight path.

The longitudinal axis of the aircraft will align with the relative wind which results in a change in the heading of the aircraft.

If you INCREASE the speed of the aircraft in a level, constant banked turn, you will increase the forward force while keeping the horizontal force relatively the same. This will cause the flightpath to be more straight off the nose. In other words, moving faster will re-vector the relative wind closer to the longitudinal axis of the aircraft. This will slow the rate of turn and increase the turn radius.

Am I on the right track here?