We report the confirmation of three transiting exoplanets orbiting TOI-1243 (LSPM J0902+7138), TOI-4529 (G 2-21), and TOI-5388 (Wolf 346) that were 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 precision of <7%, the mass of TOI-1243 b with a precision of 19%, and upper mass limits for TOI-4529 b and TOI-5388 b. The radius of TOI-1243 b is 2.33 ± 0.12 R⊕, its mass is 7.7 ± 1.5 M⊕, and the mean density is 0.61 ± 0.15 ρ⊕. The radius of TOI-4529b is 1.77−0.08+0.09 R⊕, the 3σ upper mass limit is 4.9 M⊕, and the 3σ upper density limit is 0.88 ρ⊕. The third planet, TOI-5388 b, is Earth-sized with a radius of 0.99−0.06+0.07 R⊕, a 3σ upper mass limit of 2.2 M⊕, and a 3σ upper density limit of 2.2 ρ⊕. 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 toward 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 in determining their real composition. The sample of small planets around M dwarfs is widely used to understand planet formation and composition theories, and our study adds three planets to this sample.