  Topological Josephson junctions designed on the surface of a 3D-topological
insulator (TI) harbor Majorana bound states (MBS's) among a continuum of
conventional Andreev bound states. The distinct feature of these MBS's lies in
the $4\pi$-periodicity of their energy-phase relation that yields a fractional
ac Josephson effect and a suppression of odd Shapiro steps under $r\!f$
irradiation. Yet, recent experiments showed that a few, or only the first, odd
Shapiro steps are missing, casting doubts on the interpretation. Here, we show
that Josephson junctions tailored on the large bandgap 3D TI Bi$_2$Se$_3$
exhibit a fractional ac Josephson effect acting on the first Shapiro step only.
With a modified resistively shunted junction model, we demonstrate that the
resilience of higher order odd Shapiro steps can be accounted for by thermal
poisoning driven by Joule overheating. Furthermore, we uncover a residual
supercurrent at the nodes between Shapiro lobes, which provides a direct and
novel signature of the current carried by the MBS. Our findings showcase the
crucial role of thermal effects in topological Josephson junctions and lend
support to the Majorana origin of the partial suppression of odd Shapiro steps.
