The exception is the poles, where no centripetal force is required because there is no tangential speed there. The centripetal force that acts on objects at the earth's surface is the full force of gravity only at the equator. The rotational inertia of you and the rotating turntable is least when you are at the rotational axis. In accord with the conservation of angular momentum, as you crawl toward the outer rim, you increase the rotational inertia of the spinning system and decrease the angular speed. It is your rotational speed that increases, not your angular momentum.
Angular momentum remains the same.
Earth that washes down the river is being deposited at a greater distance from the earth's rotational axis. Just as the man on the turntable slows down when one of the masses is extended, the earth slows down in its rotational motion, extending the length of the day. The amount of slowing, of course, is exceedingly small. In accord with the conservation of angular momentum, as the radial distance of mass increases, the angular speed decreases. The mass of material used to construct skyscrapers is lifted, slightly increasing the radial distance from the earth's spin axis.