We report a detailed study of the transport coefficients of $beta$-Bi$_4$I$_4$ quasi-one dimensional topological insulator. Electrical resistivity, thermoelectric power, thermal conductivity and Hall coefficient measurements are consistent with the possible appearance of a charge density wave order at low temperatures. Both electrons and holes contribute to the conduction in $beta$-Bi$_4$I$_4$ and the dominant type of charge carrier changes with temperature as a consequence of temperature-dependent carrier densities and mobilities. Measurements of resistivity and Seebeck coefficient under hydrostatic pressure up to 2 GPa show a shift of the charge density wave order to higher temperatures suggesting a strongly one-dimensional character at ambient pressure. Surprisingly, superconductivity is induced in $beta$-Bi$_4$I$_4$ above 10 GPa with of 4.0 K which is slightly decreasing upon increasing the pressure up to 20 GPa. Chemical characterisation of the pressure-treated samples shows amorphization of $beta$-Bi$_4$I$_4$ under pressure and rules out decomposition into Bi and BiI$_3$ at room-temperature conditions.
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