  We investigate strong coupling topological superconductivity in twisted
moir\'e bilayer WSe$_2$. Our approach is based on an effective $t$-$J$ model
with displacement-field-dependent complex hoppings, which is treated with the
variational Gutzwiller projection method. The calculated phase diagram contains
domes of topologically nontrivial superconducting phases, with Chern numbers
$C=\pm 2,\;\pm 4$. The order parameter is characterized by a mixed
$d$+$id$-wave (singlet) and $p$-$ip$-wave (triplet) gap symmetry. We also
report on the appearance of an additional topologically trivial extended
$s$-wave and $f$-wave paired phase. As we show, by changing the electron
density and displacement field, one can tune the singlet and triplet
contributions to the pairing, as well as induce topological phase transitions
between superconducting states characterized by different values of the Chern
number. We analyze the physical origin of the reported effects and discuss it
briefly in the view of new possibilities for designing unconventional
superconductivity in moir\'e systems.
