ABSTRACT We present the discovery and characterization of two transiting planets observed by TESS in the light curves of the young and bright (V = 9.67) star HD73583 (TOI-560). We perform an intensive spectroscopic and photometric space- and ground-based follow-up in order to confirm and characterize the system. We found that HD73583 is a young (∼500 Myr) active star with a rotational period of 12.08 ± 0.11  d, and a mass and radius of 0.73 ± 0.02 M⊙ and 0.65 ± 0.02 R⊙, respectively. HD 73583 b (Pb = $6.3980420 _{ - 0.0000062 } ^ { + 0.0000067 }$ d) has a mass and radius of $10.2 _{ - 3.1 } ^ { + 3.4 }$ M⊕ and 2.79 ± 0.10 R⊕, respectively, which gives a density of $2.58 _{ - 0.81 } ^ { + 0.95 }$ ${\rm g\, cm^{-3}}$. HD 73583 c (Pc = $18.87974 _{ - 0.00074 } ^ { + 0.00086 }$ d) has a mass and radius of $9.7 _{ - 1.7 } ^ { + 1.8 }$ M⊕ and $2.39 _{ - 0.09 } ^ { + 0.10 }$ R⊕, respectively, which translates to a density of $3.88 _{ - 0.80 } ^ { + 0.91 }$ ${\rm g\, cm^{-3}}$. Both planets are consistent with worlds made of a solid core surrounded by a volatile envelope. Because of their youth and host star brightness, they both are excellent candidates to perform transmission spectroscopy studies. We expect ongoing atmospheric mass-loss for both planets caused by stellar irradiation. We estimate that the detection of evaporating signatures on H and He would be challenging, but doable with present and future instruments.