Recent observational analyzes have revealed a significant tension in the growth index $\gamma$, which characterizes the growth rate of cosmic structures. When $\gamma$ is treated as a free parameter within $\Lambda$CDM, the PL18+$f{\sigma }_8$ combination yields $\gamma \approx 0.64$, in $\sim 4\sigma$ tension with the theoretically expected value $\gamma \approx 0.55$ (assuming general relativity), suggesting a preference for suppressed late-time growth relative to the standard expectation. In this work, we investigate whether the $\Lambda_{\rm s}$CDM framework —featuring a rapid sign-switching cosmological constant (mirror AdS-to-dS transition) in the late universe at redshift $z_{†}\sim 2$— can alleviate the $\gamma$ tension together with the $H_0$ and $S_8$ tensions. We find that $\gamma\Lambda_{\rm s}$CDM is able to reduce the $\gamma$ and $H_0$ tensions relative to $\gamma \Lambda$CDM while maintaining low $S_8$ values. For PL18+$f{\sigma }_8$, the combination underlying the $\gamma$ tension, $\gamma\Lambda_{\rm s}$CDM yields $z_{†}>1.58$ (95% C.L.) and reduces the $\gamma$ and $H_0$ discrepancies from $3.7\sigma$ and $4.2\sigma$ in $\gamma \Lambda$CDM to $2.1\sigma$ and $1.1\sigma$, respectively, while leaving $\chi^2_{\rm min}$ essentially unchanged. The largest improvement among the free-$z_{†}$ analyses is obtained for PL18+$f{\sigma }_8$+PP&SH0ES, for which the marginalized posterior constraint is $z_{†}={1.82}_{-0.28}^{+0.32}$ (95% C.L.); the $\gamma$ and $H_0$ tensions are reduced to $1.8\sigma$ and $0.6\sigma$, respectively, and the model also provides an improved fit to the data. The efficacy of the mechanism is, however, dataset dependent: the standard 3D BAO measurements used in the present analysis —from SDSS-IV/eBOSS and the first year of DESI observations— tend to push the transition to higher redshifts, $z_{†}\gtrsim 2.3$–$2.4$, and weaken the simultaneous alleviation, particularly of the $H_0$ tension. We also examine the benchmark case with fixed $z_{†}=1.7$, previously identified in studies of the standard ($\gamma \simeq 0.55$) $\Lambda_{\rm s}$CDM framework as a redshift of interest for alleviating multiple major cosmological tensions —including those in $H_0$, $M_{\rm B}$, and $S_8$. For the restricted combinations in which it is considered, this benchmark yields low residual tensions (e.g., a $1.6\sigma$ $\gamma$ tension and an effectively vanishing $H_0$ tension for PL18+$f{\sigma }_8$), albeit with somewhat worse fits than $\gamma \Lambda$CDM. It is, however, excluded at more than $2\sigma$ by the free-$z_{†}$ analyses once either the uncalibrated PantheonPlus SN~Ia sample or the 3D BAO data are included. Our results indicate that $\Lambda_{\rm s}$CDM can contribute to the joint alleviation of multiple cosmological tensions, although its impact is dataset dependent.