  We here report a detailed high-pressure infrared transmission study of BiTeCl
and BiTeBr. We follow the evolution of two band transitions: the optical
excitation $\beta$ between two Rashba-split conduction bands, and the
absorption $\gamma$ across the band gap. In the low pressure range, $p< 4$~GPa,
for both compounds $\beta$ is approximately constant with pressure and $\gamma$
decreases, in agreement with band structure calculations. In BiTeCl, a clear
pressure-induced phase transition at 6~GPa leads to a different ground state.
For BiTeBr, the pressure evolution is more subtle, and we discuss the
possibility of closing and reopening of the band gap. Our data is consistent
with a Weyl phase in BiTeBr at 5$-$6~GPa, followed by the onset of a structural
phase transition at 7~GPa.
