It has recently become possible to chemically synthesize atomically flat semiconductor nanoplatelets with monolayer-precision control over the platelet thickness. It has been suggested that these platelets are quantum wells; that is, carriers in these platelets are confined in one dimension but are free to move in the other two dimensions. Here, we report time-resolved photoluminescence and transient-absorption measurements of carrier relaxation that confirm the quantum-well nature of these nanomaterials. Excitation of the nanoplatelets by an intense laser pulse results in the formation of a high-temperature carrier population that cools back down to ambient temperature on the time scale of several picoseconds. The rapid carrier cooling indicates that the platelets are well-suited for optoelectronic applications such as lasers and modulators.