Is geodesic distance equivalent to "norm distance" in $SL_n(\mathbb{R})$?
Here's what I see in Einsiedler-Ward:
Your norm on $SL_n(\mathbb{R})$ is induced by the operator norm on the vector space $M_{n}(\mathbb{R})$. Being a norm on a finite-dimensional vector space, it is equivalent to the euclidean norm $\|\cdot\|$ on $M_{n}(\mathbb{R})$.
Now any left-invariant Riemannian metric $\langle \ ,\rangle$ on $TG=G\times \mathfrak{g}$ is determined by its restriction to $TG_{I}=\{I\}\times \mathfrak{g} \cong \mathfrak{g} \subset M_n(\mathbb{R})$. Without loss of generality, we can assume that the Riemannian metric restricted to $\mathfrak{g}$ is induced by the euclidean norm on $M_n(\mathbb{R}).$ Let $d$ be the distance on $G$ induced by the path integral formula of $\langle \ , \rangle$.
Let $B$ be a pre-compact neighbourhood of $I$ in $G$ where the local inverse ($\log$) of the exponential map is defined. Assume $\log(B)$ is a convex ball in $\mathfrak{g}$. Let $B'$ be another pre-compact neighbourhood containing the closure of $B$.
Say $\phi:[0,1] \to B'$ joins $g_0, g_1 \in B$. Then since the norm of $\phi(t)$ is bounded, we get $c>0$ (independent of $g_0,g_1$) such that
$$L(\phi):= \int\langle D\phi(t), D\phi(t) \rangle^{1/2}dt = \int \left\langle DL^{-1}_{\phi(t)}\circ D\phi(t), DL^{-1}_{\phi(t)}\circ D\phi(t)\right\rangle^{1/2} dt = \int \|\phi(t)^{-1}\phi'(t)\| dt \\ \geq c\int\|\phi'(t)\|dt \geq c\| g_1-g_0\|.$$
This shows that $c\|g_1-g_0\| \leq d(g_0,g_1)$ if the infimum of path integrals is taken over paths which remain in $B'$. But since $d\left(B,(B')^c\right)>0,$ we can assume that this estimate holds in general. Hence for all $g_0,g_1 \in B$, we have
\begin{equation} c\|g_1-g_0\| \leq d(g_0,g_1) \qquad (1) \end{equation}
and it remains to show a reverse inequality.
Consider the path $\phi:[0,1] \to B$ given by $t \mapsto \exp\left(\log g_0 + t(\log g_1-\log g_0)\right)$. This is well defined since we assumed $\log(B)$ was a convex ball in $\mathfrak{g}$. Then, since the norm of $\phi(t)^{-1}$ is bounded, and since $d(\exp)$ is bounded in $\log(B)$ and since $\log$ is Lipschitz (by the mean value theorem) in a neighbourhood of $I$,
$$d(g_0,g_1) \leq \int\langle D\phi(t), D\phi(t) \rangle^{1/2}dt = \int \|\phi(t)^{-1}\phi'(t)\|dt \leq \int C_1\|\phi'(t)\|dt \leq C_1C_2\|g_1-g_0\|$$
for some $C_1, C_2>0$ (independent of $g_0, g_1$). Hence for all $g_0,g_1\in B$, we have
$$ d(g_0,g_1) \leq C \|g_0-g_1\|. \qquad (2)$$