Aims. We present the atmospheric structure and the fundamental properties of the red supergiants (RSGs) AH Sco, UY Sct, and KW Sgr based on VLTI/AMBER observations. Methods. We carried out spectro-interferometric observations of AH Sco, UY Sct, and KW Sgr in the near-infrared K band (1.92-2.47 μm) with the VLTI/AMBER instrument with spatial and spectral resolutions of 3 milliarcsec and 1500, respectively, and compared the data to a new grid of hydrostatic PHOENIX model atmospheres.Results. In our visibility data, we observe molecular layers of water and CO in extended atmospheres. For a uniform disk modeling, we observe size increases at the water band centered at 1.9 μm of 10% to 25% and at the CO bandheads at 2.3-2.5 μm of 20%-35% with respect to the near-continuum bandpass at around 2.20 μm. Our near-infrared spectra of AH Sco, UY Sct, and KW Sgr are well reproduced by the PHOENIX model atmospheres. The continuum visibility values are consistent with a limb-darkened disk as predicted by the PHOENIX models. However, the model visibilities do not predict the large observed extensions of the molecular layers. Comparing the continuum visibility values to PHOENIX models, we estimate the Rosseland-mean photospheric angular diameters of AH Sco, UY Sct, and KW Sgr to be 5.81 ± 0.15 mas, 5.48 ± 0.10 mas, and 3.91 ± 0.25 mas, respectively. Together with the distance and the spectro-photometry, we calculate radii of 1411 ± 124 R for AH Sco, 1708 ± 192 R for UY Sct, and 1009 ± 142 R for KW Sgr and effective temperatures of 3682 ± 190 K for AH Sco, 3365 ± 134 K for UY Sct, and 3720 ± 183 K for KW Sgr. Conclusions. AH Sco, UY Sct, and KW Sgr exhibit extended atmospheric layers of H 2 O and CO. The PHOENIX atmosphere models predict the spectra and the continuum visibility values, but cannot reproduce the large extensions of the molecular layers. This indicates that the opacities of the molecular bands are included, but that the model atmospheres are too compact compared to the observations. The observed extended layers may be levitated by processes such as pulsation or convection, which are not included in the hydrostatic atmospheric models. The location of the targets in the HR-diagram is confirmed to be close to, and possibly slightly to the right of, the Hayashi limit of recent evolutionary tracks corresponding to masses between about 20 M and 40 M .