The structure of $^{34}$Si was studied through $gamma$ spectroscopy separately in the $beta^-$ decays of $^{34}$Mg and $^{34}$Al at the ISOLDE facility of CERN. Different configurations in $^{34}$Si were populated independently from the two recently identified $beta$-decaying states in $^{34}$Al having spin-parity assignments $J^pi = 4^-$ dominated by the normal configuration $pi (d_{5/2})^{-1} otimes u (f_{7/2})$ and $J^pi = 1^+$ by the intruder configuration $pi (d_{5/2})^{-1} otimes u (d_{3/2})^{-1}(f_{7/2})^{2}$. The paper reports on spectroscopic properties of $^{34}$Si such as an extended level scheme, spin and parity assignments based on log($ft$) values and $gamma$-ray branching ratios, absolute $beta$ feeding intensities and neutron emission probabilities. A total of 11 newly identified levels and 26 transitions were added to the previously known level scheme of $^{34}$Si. Large scale shell-model calculations using the {sc sdpf-u-mix} interaction, able to treat higher order intruder configurations, are compared with the new results and conclusions are drawn concerning the predictive power of {sc sdpf-u-mix}, the $N=20$ shell gap, the level of mixing between normal and intruder configurations for the 0$_1^+$, 0$_2^+$ and 2$_1^+$ states and the absence of triaxial deformation in $^{34}$Si.
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