  Two-dimensional electron gases with strong spin-orbit coupling covered by a
superconducting layer offer a flexible and potentially scalable platform for
Majorana networks. We predict Majorana bound states (MBSs) to appear for
experimentally achievable parameters and realistic gate potentials in two
designs: either underneath a narrow stripe of a superconducting layer
(S-stripes) or where a narrow stripe has been removed from a uniform layer
(N-stripes). The coupling of the MBSs can be tuned for both types in a wide
range (< 1 neV to >10 $\mu$eV) using gates placed adjacent to the stripes. For
both types, we numerically compute the local density of states for two parallel
Majorana-stripe ends as well as Majorana trijunctions formed in a tuning-fork
geometry. The MBS coupling between parallel Majorana stripes can be suppressed
below 1 neV for potential barriers in the meV range for separations of about
200 nm. We further show that the MBS couplings in a trijunction can be
gate-controlled in a range similar to the intra-stripe coupling while
maintaining a sizable gap to the excited states (tens of $\mu$eV). Altogether,
this suggests that braiding can carried out on a time scale of 10-100 ns.
