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How is it that contact lenses can be thinner than bifocales but just as effective at fixing someone's sight?

The power or 1/f of contact or framed glasses for the same person needs to be the same. Power is proportional to (N-1)[1/R1 - 1/R2]. N is the index of refraction of the material. N is not much different for contact or glass lenses. The difference in curvatures, [1/R1 - 1R2], is therefore also about the same

The reasons glass lenses are thinner are twofold:
(1) Glass lenses maintain an edge thickness of about 1/8" to make them less fragile and easier for the frames to hold.
(2) Contact lenses only have to cover the pupil of the eye, which means that their diameter can be as small as 1 cm. Glass lenses need to have at least 3 cm clear aperture because thay are located farther fcrom the eye and need to provide a wide field of view.
(3) The increase of thickness at the center, relative to the edge, is proportional to the square of the aperture size. If the glass lens has (typically) three times the diameter of the contact lens, its thickness change at the center must be nine times larger. Add the increased edge thickness to this and you get a much thicker glass lens.

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