  Inspired by recent experimental measurements [Guo \textit{et al.}, Phys. Rev.
Lett.~\textbf{124}, 206602 (2020); Jim\'enez \textit{et al.}, Nature
\textbf{592}, 370 (2021)] on frustrated quantum magnet SrCu$_2$(BO$_3$)$_2$
under combined pressure and magnetic fields, we study the related spin-$1/2$
Shastry-Sutherland (SS) model using state-of-the-art tensor network methods. By
calculating thermodynamics, correlations and susceptibilities, we find, in zero
magnetic field, not only a line of first-order plaquette-singlet (PS) to
dimer-singlet phase transition ending with a critical point, but also
signatures of the ordered PS transition with its critical endpoint terminating
on this first-order line. Moreover, we uncover prominent magnetic barocaloric
responses, a novel type of quantum correlation induced cooling effect, in the
strongly fluctuating supercritical regime. Under finite fields, we identify a
quantum phase transition from the PS phase to the spin supersolid phase that
breaks simultaneously lattice translational and spin rotational symmetries. The
present findings on the SS model are accessible in current experiments and
would shed new light on exotic critical and supercritical phenomena in
archetypal frustrated quantum magnets.
