When the front wheel is suddenly turned, a large frictional force arises
at the tire/road interface, directed in the present -v direction (toward
the rear). Resolve this large force into two components, tangential and
perpendicular to the TURNED wheel. The tangential component will produce
a torque and rotationally accelerate the wheel in its new direction. The
perpendicular component is "thwarted" by static friction; the wheel's
response to this large "irresistible" perpendicular force is tantamount to
the wheel "pushing itself off a wall". This push is directed "inward",
perpendicular to the (turned) wheel plane and is, I think, the major
"source" of the centripetal force which Ludwik is seeking.