We report the confirmation of three transiting exoplanets orbiting TOI-1243 (LSPM J0902+7138), TOI-4529 (G 2-21) and TOI-5388 (Wolf 346) initially detected by TESS, through ground-based photometry and radial-velocity follow-up measurements with CARMENES. The planets present short orbital periods of 4.65, 5.88, and 2.59 days, and they orbit early M dwarfs (M2.0 V, M1.5 V, and M3.0 V, respectively). We were able to precisely determine the radius of all three planets with a <7% precision, the mass of TOI-1243b with 19% precision and upper mass limits for TOI-4529b and TOI-5388b. TOI-1243b has a radius of 2.33+/-0.12R_{Earth}_, a mass of 7.7+/-1.5M_{Earth}_, and a mean density of 0.61+/-0.15{rho}_{Earth}_. TOI-4529b has a radius of 1.77^+0.09^_-0.08_R_{Earth}_, a 3{sigma} upper mass limit of 4.9 M_{Earth} and a 3{sigma} upper density limit of 0.88{rho}_{Earth}_. The third planet, TOI-5388b, is Earth-sized with a radius of 0.99^+0.07^_-0.06_R_{Earth}_, a 3{sigma} upper mass limit of 2.2M_{Earth}_ and a 3{sigma} upper density limit of 2.2{rho}_{Earth}_. While TOI-5388 b is most probably rocky, given its Earth-like radius, TOI-1243 b and TOI-4529 b are located in a highly degenerate region in the mass-radius space. TOI-4529 b appears to lean towards a water-world composition. TOI-1243 b has enough mass to host a significant H-He envelope, although a water-world and pure rocky compositions are also consistent with the data. Our analysis indicates that future atmospheric observations using JWST can aid into determining their real composition. The sample of small planets around M-dwards is widely used to understand planet formation and composition theories, and our study adds 3 more planets to this sample.