The issue we are concerned with here is how we get the plane to move along a circle of radius r. Start with thinking about the plane flying with the wings level (in a horizontal plane). There are two forces acting on the plane in the vertical direction: its weight, Mg, and the "lift", L, supplied by the wings. With the plane in level flight, these vertical forces must balance.
Now we want the plane to make a turn, so we "bank" the wings by an angle Q. So the "lifting force" will be off-vertical by an angle Q; since it must still balance the weight of the plane vertically, its magnitude will change to L' . (How is it related to Mg?)
This altered lifting force now has a horizontal component as well. This component of L', which will equal L' sin Q , is what supplies the centripetal force to pull the plane into a circular path (for the interval of the wing-banking). So we have
L' sin Q = M·(v^2)/r .
Try things from there.
BTW, a similar argument can be used to explain why a cyclist (leg- or motor-powered) must "lean into the turn" when they want to go round a corner...