Pressure-induced structural changes of conducting halide perovskites $\left({\mathrm{CH}}_3{\mathrm{NH}}_3\right){\mathrm{SnI}}_3,$ $\left({\mathrm{CH}}_3{\mathrm{NH}}_3{)}_{0.5}\right({\mathrm{NH}}_2\mathrm{CH}={\mathrm{NH}}_2{)}_{0.5}{\mathrm{SnI}}_3,$ and $\left({\mathrm{NH}}_2\mathrm{CH}={\mathrm{NH}}_2\right){\mathrm{SnI}}_3,$ have been investigated using synchrotron x-ray powder diffraction. In contrast to low-temperature structural changes, no evidence of an increased ordering of the organic cations was observed under pressure. Instead, increase in pressure results first in a ${\mathrm{ReO}}_3$-type doubling of the primitive cubic unit cell, followed by a symmetry distortion, and a subsequent amorphization above 4 GPa. This process is reversible and points towards a pressure-induced templating role of the organic cation. Bulk compressions are continuous across the phase boundaries. The compressibilities identify these hybrids as the most compressible perovskite system ever reported. However, the $\mathrm{Sn}-I$ bond compressibility in $\left({\mathrm{CH}}_3{\mathrm{NH}}_3\right){\mathrm{SnI}}_3$ shows a discontinuity within the supercell phase. This is possibly due to an electronic localization.