Current observations favour that the massive ultraviolet-bright clumps with a median stellar mass of $\sim 10^7~M_{\odot}$, ubiquitously observed in $z\sim 1-3$ galaxies, are star-forming regions formed in-situ in galaxies. It has been proposed that they result from gas fragmentation due to gravitational instability of gas-rich, turbulent, high-redshift discs. We bring support to this scenario by reporting the new discovery of giant molecular clouds (GMCs) in the strongly lensed, clumpy, main-sequence galaxy, A521-sys1, at $z=1.043$. Its CO(4-3) emission was mapped with the Atacama Large Millimeter/submillimeter Array (ALMA) at an angular resolution of $0.19''\times 0.16''$, reading down to 30~pc thanks to gravitational lensing. We identified 14 GMCs, most being virialized, with $10^{5.9}- 10^{7.9}~M_{\odot}$ masses and a median $800~M_{\odot}~\rm{pc}^{-2}$ molecular gas mass surface density, that are, respectively, 100 and 10 times higher than for local GMCs. They are also characterized by 10 times higher supersonic turbulence with a median Mach number of 60. They end up to fall above the Larson scaling relations, similarly to the GMCs in another clumpy $z\simeq 1$ galaxy, the Cosmic Snake, although noteworthy differences between the two sets of high-redshift GMCs exist. Altogether they support that GMCs form with properties that adjust to the ambient interstellar medium conditions prevalent in the host galaxy whatever its redshift. The detected A521-sys1 GMCs are massive enough to be the parent gas clouds of stellar clumps, with a relatively high star-formation efficiency per free-fall time of $\sim 11$ per cent.