High-temperature superconductivity is known to involve the pairing of charge carriers, but the precise nature of these carriers and the mechanism of their pairing remain unclear, The copper oxides are known to exhibit a strong Jahn-Teller effect-in which spontaneous lattice distortions remove the degeneracy of the electronic ground state - and it has been suggested that the charge carriers are Jahn-Teller polarons (bare charge carriers accompanied by local lattice distortions). In fact, the demonstration(1) that a strong Jahn-Teller effect can lead to the formation of such polarons led to the original discovery of high-temperature superconductivity(2). Still, direct evidence that Jahn-Teller polarons exist in the superconducting state of the copper oxides has been lacking, although some indirect evidence comes from their recent discovery(3) in the structurally similar but non-superconducting manganite La1-xCaxMnO3. Here we report the results of magnetization and thermal expansion measurements on samples of the copper oxide superconductor La2-xSrxCuO4 which characterize the oxygen-isotope effects on the carrier density and on the in-plane penetration depth. We find a negligible isotope effect on the former, but a large effect on the latter, Specific quantitative features of the results show that polaronic charge carriers exist and condense into Cooper pairs in the copper oxide superconductors.