We present an extensive CO emission-line survey of the Planck’s dusty Gravitationally Enhanced subMillimetre Sources, a small set of 11 strongly lensed dusty star-forming galaxies at z = 2–4 discovered with Planck and Herschel satellites, using EMIR on the IRAM 30-m telescope. We detected a total of 45 CO rotational lines from J_up = 3 to J_up = 11, and up to eight transitions per source, allowing a detailed analysis of the gas excitation and interstellar medium conditions within these extremely bright (μL_FIR = 0.5 − 3.0 × 10¹⁴L_⊙), vigorous starbursts. The peak of the CO spectral-line energy distributions (SLEDs) fall between J_up = 4 and J_up = 7 for nine out of 11 sources, in the same range as other lensed and unlensed submillimeter galaxies (SMGs) and the inner regions of local starbursts. We applied radiative transfer models using the large velocity gradient approach to infer the spatially-averaged molecular gas densities, n_H2 ≃ 10^2.6 − 10^4.1 cm⁻³, and kinetic temperatures, T_k ≃ 30–1000 K. In five sources, we find evidence of two distinct gas phases with different properties and model their CO SLED with two excitation components. The warm (70–320 K) and dense gas reservoirs in these galaxies are highly excited, while the cooler (15–60 K) and more extended low-excitation components cover a range of gas densities. In two sources, the latter is associated with diffuse Milky Way-like gas phases of density n_H2 ≃ 10^2.4 − 10^2.8 cm⁻³, which provides evidence that a significant fraction of the total gas masses of dusty starburst galaxies can be embedded in cool, low-density reservoirs. The delensed masses of the warm star-forming molecular gas range from 0.6to12 × 10¹⁰ M_⊙. Finally, we show that the CO line luminosity ratios are consistent with those predicted by models of photon-dominated regions (PDRs) and disfavor scenarios of gas clouds irradiated by intense X-ray fields from active galactic nuclei. By combining CO, [C I] and [C II] line diagnostics, we obtain average PDR gas densities significantly higher than in normal star-forming galaxies at low-redshift, as well as far-ultraviolet radiation fields 10²–10⁴ times more intense than in the Milky Way. These spatially-averaged conditions are consistent with those in high-redshift SMGs and in a range of low-redshift environments, from the central regions of ultra-luminous infrared galaxies and bluer starbursts to Galactic giant molecular clouds.