  Background: Electric-quadrupole ($E2$) strengths relate to the underlying
quadrupole deformation of a nucleus and present a challenge for many nuclear
theories. Mirror nuclei in the vicinity of the line of $N=Z$ represent a
convenient laboratory for testing deficiencies in such models, making use of
the isospin-symmetry of the systems. Purpose: Uncertainties associated with
literature $E2$ strengths in \textsuperscript{23}Mg are some of the largest in
$T_z=\left|\frac{1}{2}\right|$ nuclei in the $sd$-shell. The purpose of the
present work is to improve the precision with which these values are known, to
enable better comparison with theoretical models. Methods: Coulomb-excitation
measurements of $^{23}$Mg and $^{23}$Na were performed at the TRIUMF-ISAC
facility using the TIGRESS spectrometer. They were used to determine the $E2$
matrix elements of mixed $E2$/$M1$ transitions. Results: Reduced $E2$
transition strengths, $B(E2)$, were extracted for \textsuperscript{23}Mg and
\textsuperscript{23}Na. Their precision was improved by factors of
approximately six for both isotopes, while agreeing within uncertainties with
previous measurements. Conclusions: A comparison was made with both shell-model
and {\it ab initio} valence-space in-medium similarity renormalization group
calculations. Valence-space in-medium similarity-renormalization-group
calculations were found to underpredict the absolute $E2$ strength - in
agreement with previous studies.
