Lifetimes of $2^+_1$ and $4^+_1$ states, as well as some negative-parity and non-yrast states, in $^{188-200}$Hg were measured using $gamma-gamma$ electronic fast timing techniques with the LaBr$_3$(Ce) detector array of the GRIFFIN spectrometer. The excited states were populated in the $epsilon/beta^+$-decay of $J^pi =7^+/2^-$ $^{188-200}$Tl produced at the TRIUMF-ISAC facility. The deduced B(E2) values are compared to different interacting boson model predictions. The precision achieved in this work over previous ones allows for a meaningful comparison with the different theoretical models of these transitional Hg isotopes, which confirms the onset of state mixing in $^{190}$Hg.
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