Dusty star-forming galaxies at high redshift (1 < z < 3) represent the most intense star-forming regions in the universe. Key aspects to these processes are the gas heating and cooling mechanisms, and although it is well known that these galaxies are gas-rich, little is known about the gas excitation conditions. Only a few detailed radiative transfer studies have been carried out owing to a lack of multiple line detections per galaxy. Here we examine these processes in a sample of 24 strongly lensed star-forming galaxies identified by the Planck satellite (LPs) at z ∼ 1.1-3.5. We analyze 162 CO rotational transitions (ranging from Jup = 1 to 12) and 37 atomic carbon fine-structure lines (C i) in order to characterize the physical conditions of the gas in the sample of LPs. We simultaneously fit the CO and C i lines and the dust continuum emission, using two different non-LTE, radiative transfer models. The first model represents a two-component gas density, while the second assumes a turbulence-driven lognormal gas density distribution. These LPs are among the most gas-rich, IR-luminous galaxies ever observed ( L L IR ( 8 − 1000 m ) ∼ 10 13 − 14.6 L ; 〈 LMISM 〉 = (2.7 1.2) × 1012 M , with L ∼ 10-30 the average lens magnification factor). Our results suggest that the turbulent interstellar medium present in the LPs can be well characterized by a high turbulent velocity dispersion ( 〈 ΔVturb 〉 ∼ 100 km s−1) and ratios of gas kinetic temperature to dust temperature 〈 Tkin/Td 〉 ∼ 2.5, sustained on scales larger than a few kiloparsecs. We speculate that the average surface density of the molecular gas mass and IR luminosity, M ISM ∼ 103-4 M pc−2 and L IR ∼ 1011-12 L kpc−2, arise from both stellar mechanical feedback and a steady momentum injection from the accretion of intergalactic gas.