Context. Although the number of exoplanets reported in the literature exceeds 5000 so far, only a few dozen of them are young planets (≤900 Myr). However, a complete characterization of these young planets is key to understanding the current properties of the entire population. Hence, it is necessary to constrain the planetary formation processes and the timescales of dynamical evolution by measuring the masses of exoplanets transiting young stars. Aims. We characterize and measure the masses of two transiting planets orbiting the 400 Myr old solar-type star HD 63433, which is a member of the Ursa Major moving group. Methods. We analysed precise photometric light curves of five sectors of the TESS mission with a baseline of ~750 days and obtained ~150 precise radial velocity measurements with the visible and infrared arms of the CARMENES instrument at the Calar Alto 3.5 m telescope in two different campaigns of ~500 days. We performed a combined photometric and spectroscopic analysis to retrieve the planetary properties of two young planets. The strong stellar activity signal was modelled by Gaussian regression processes. Results. We have updated the transit parameters of HD 63433 b and c and obtained planet radii of R_p^b = 2.140 ± 0.087 R_⊕ and R_p^c = 2.692 ± 0.108 R_⊕. Our analysis allowed us to determine the dynamical mass of the outer planet with a 4σ significance (M_p^c = 15.54 ± 3.86 M_⊕) and set an upper limit on the mass of the inner planet at 3σ (M_p^b < 21.76 M_⊕). According to theoretical models, both planets are expected to be sub-Neptunes, whose interiors mostly consist of silicates and water with no dominant composition of iron, and whose gas envelopes are lower than 2% in the case of HD 63433 c. The envelope is unconstrained in HD 63433 b.