  We study the electronic and topological properties of fully relaxed twisted
bilayer (TBG) and double bilayer (TDBG) graphene under perpendicular pressure.
An approach has been proposed to obtain the equilibrium in-plane structural
deformation and out-of-plane corrugation in moir\'{e} superlattices under
pressure. We find that the in-plane relaxation becomes much stronger under
higher pressure, while the corrugation height in each layer is maintained. The
comparison between band structures of relaxed and rigid structures demonstrates
that not only the gaps on the electron and hole sides ($\Delta_e$ and
$\Delta_h$) are significantly underestimated without relaxation but also the
detailed dispersions of the middle bands of rigid structures are rather
different from those of relaxed systems. $\Delta_e$ and $\Delta_h$ in TBG reach
maximum values around critical pressures with narrowest middle bands.
Topological transitions occur in TDBG under pressure with the middle valence
and conduction bands in one valley touching and their Chern numbers transferred
to each other. The pressure can also tune the gap at the neutrality point of
TDBG, which becomes closed for a pressure range and reopened under higher
pressure. The behavior of electronic structure of supertlattices under pressure
is sensitive to the twist angle $\theta$ with the critical pressures generally
increase with $\theta$.
