We present the discovery and characterisation of two transiting planets observed by \textit{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 characterise the system. We found that HD73583 is a young (\(\sim 500\)~Myr) active star with a rotational period of \(12.08 \pm 0.11 \)\,d, and a mass and radius of \( 0.73 \pm 0.02 M_\odot\) and \(0.65 \pm 0.02 R_\odot\), respectively. HD73583 b (\(P_b=6.3980420 _{ - 0.0000062 }^{+0.0000067}\) d) has a mass and radius of \(10.2 _{-3.1}^{+3.4} M_\oplus\) and\(2.79 \pm 0.10 R_\oplus\), respectively, that gives a density of \(2.58 _{-0.81}^{ 0.95} {\rm g\,cm^{-3}}\). HD73583 c (\(P_c= 18.87974 _{-0.00074 }^{+0.00086}\)) has a mass and radius of \(9.7_{-1.7} ^ {+1.8} M_\oplus\) and \(2.39_{-0.09}^{+0.10} R_\oplus\), respectively, this 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.