$\newcommand{\al}{\alpha}$You are citing Szegő's book incorrectly. The correct citation is this: for every real $\al$ and every real $a>0$, 
\begin{equation*}
	\max_{x\ge a}e^{-x/2} x^{\al/2+1/4}|L_n^{(\al)}(x)| \sim n^{\al/2-1/12} \tag{1}\label{1}
\end{equation*}
as $n\to\infty$. 

There can be no corresponding result with $\min$ in place of $\max$ -- because, for large enough $n$, the polynomial $L_n^{(\al)}$ oscillates, so that for every real $\al$, every real $a>0$, and all large enough $n$  
\begin{equation*}
	\min_{x\ge a}e^{-x/2} x^{\al/2+1/4}|L_n^{(\al)}(x)| =0. \tag{2}\label{2}
\end{equation*}

Indeed, by Theorem 6.31.3 (p. 129) of G. Szegő, Orthogonal polynomials, 1939, for $\al>-1$ and the $\nu$th smallest root $x_{\nu n}$ of $L_n^{(\al)}(x)$ we have 
\begin{equation*}
	x_{\nu n}>\frac{(j_\nu/2)^2}{n+(\al+1)/2}
\end{equation*}
where $\nu=1,\dots,n$ and, according to the first sentence after that theorem, 
$(j_\nu/2)^2\sim\pi^2\nu^2/4$ as $\nu\to\infty$. So, for any fixed natural $k$, uniformly over  all $\nu\in\{n-k+1,\dots,n\}$ we have $x_{\nu n}\gtrsim\frac{\pi^2}4\,n$. 

Therefore and because $\frac{\pi^2}4>1$, for any real $\al>-1$, any fixed natural $k$, and all large enough $n$, there are at least $k$ roots of $L_n^{(\al)}(x)$ in the interval $[n,\infty)$, and hence \eqref{2} holds. 

By Rolle's theorem and the formula 
\begin{equation*}
	\frac d{dx}\,[x^\al e^{-x}\,L_n^{(\al)}(x)]=(n+1)x^{\al-1} e^{-x}\,L_{n+1}^{(\al-1)}(x) 
\end{equation*}
at the [top of page 287][2], it follows that for any fixed natural $k$, any real $\al>-1$, and all large enough $n$, there are at least $k-1$ roots of $L_{n+1}^{(\al-1)}(x)$ in the interval $[n,\infty)$. That is, for any fixed natural $k$, any real $\al>-1-1$, and all large enough $n$, there are at least $k-1$ roots of $L_n^{(\al)}(x)$ in the interval $[n-1,\infty)$. Continuing thus, we conclude that, for any fixed natural $k$, any real $\al>-1-(k-1)$, and all large enough $n$, there is at least one root of $L_n^{(\al)}(x)$ in the interval $[n-k,\infty)$. 
That is, for any real $\al$, any fixed natural $k>-\al$, and all large enough $n$, there is at least one root of $L_n^{(\al)}(x)$ in the interval $[n-k,\infty)$.  

Thus, we have \eqref{2} for every real $\al$, every real $a>0$, and all large enough $n$. 

[1]: https://en.wikipedia.org/wiki/Laguerre_polynomials#Explicit_examples_and_properties_of_the_generalized_Laguerre_polynomials 
[2]: https://www.cambridge.org/core/books/special-functions/1F1C575CEA780EE774F5518C7963BF08