We will argue by induction on the dimension of $\mathscr{X}$. The case $\dim(\mathscr{X})=1$ follows at once from the fact that all strata of dlt pairs are normal [Ko13, 4.16(2)]. Thus we may assume that $\dim(\mathscr{X})\geq 2$, and that the result holds for pairs of strictly lower dimension.

Let $x$ be a point on $C$. We claim that every prime divisor in $D$ that contains $C$ is Cartier at $x$. Assuming the claim for now, it follows that $C$ is a local complete intersection at $x$, and thus reduced because it is generically reduced (the pair $(\mathscr{X},D)$ is snc at the generic point of $C$). Now it follows from [Ko13, 4.16(2)] that $C$ is normal, and thus regular since it is of dimension one. But $C$ is defined by the local equations at $x$ of the prime components of $D$ that contain $C$; these local equations form a regular sequence, again by [Ko13, 4.16(2)]. We conclude that locally at $x$, the scheme $\mathscr{X}$ is regular and $D$ is a strict normal crossings divisor.

Thus it suffices to prove our claim. We may assume that $x$ is not a zero-dimensional stratum of $D$, since at such points, the pair $(\mathscr{X},D)$ is snc by the definition of a dlt pair. Let $E$ be a prime divisor in $D$ that contains $C$. Let $F_{1},\ldots,F_{r}$ be the non-empty intersections of $E$ with the other components of $D$, and set $\Delta=F_{1}+\ldots+F_{r}$. Then the pair $(E,\Delta)$ is dlt, and

is Cartier (see Proposition 4.5 and Claim 4.16.4 in [Ko13]). By the induction hypothesis, $E$ is regular at $x$.

Let $m\geq 1$ be the index of $E$ at $x$, that is, the smallest positive integer such that $mE$ is Cartier at $x$. Working locally around $x$, we may assume that $E$ is regular and that $\mathcal{O}_{\mathscr{X}}(mE)$ is a trivial line bundle. The choice of a trivialisation determines a ramified $\mu_{m}$-cover $h\colon\widetilde{\mathscr{X}}\to\mathscr{X}$ defined by

Here $\mathcal{O}_{\mathscr{X}}(-aE)$ is the rank one reflexive sheaf associated with the Weil divisor $-aE$. This is the so-called index one cover of the pair $(\mathscr{X},E)$ at the point $x$; see [KM98, 2.52] for details. The morphism $h$ is étale over all the points where $E$ is Cartier; in particular, it is étale over all the codimension one points of $E$, since $\mathscr{X}$ is regular in codimension two by Lemma 4.2. The minimality of $m$ implies that the inverse image of $x$ in $\widetilde{\mathscr{X}}$ consists of a unique point, which we denote by $\widetilde{x}$.

We write $\widetilde{E}$ for the inverse image of $E$ on $\widetilde{\mathscr{X}}$, and $\widetilde{D}$ for the inverse image of the divisor $D$. By [KM98, 5.20], the pair $(\widetilde{\mathscr{X}},\widetilde{D})$ is log canonical, and $\mathrm{mld}_{\widetilde{x}}(\widetilde{\mathscr{X}},\widetilde{D})$ is positive. Since we chose $x$ on a one-dimensional stratum $C$, the divisor $D$ has $\dim(\mathscr{X})-1$ irreducible components that pass through $x$. This implies that $\widetilde{E}$ is unibranch at $\widetilde{x}$. Otherwise, étale-locally around $\widetilde{x}$, the divisor $\widetilde{D}$ would have at least $\dim(\mathscr{X})$ components passing through $\widetilde{x}$, and $\widetilde{x}$ would be their intersection; but this implies that $\widetilde{x}$ is a log canonical center of $(\widetilde{X},\widetilde{D})$, by [Ko13, 4.41(2)], contradicting the positivity of $\mathrm{mld}_{\widetilde{x}}(\widetilde{\mathscr{X}},\widetilde{D})$.

We denote by $\widetilde{E}^{\prime}$ the normalization of $\widetilde{E}$. Since $\widetilde{E}$ is unibranch at $\widetilde{x}$, there is a unique point $\widetilde{x}^{\prime}$ on $\widetilde{E}^{\prime}$ that lies above $\widetilde{x}\in\widetilde{E}$. We have already observed that the morphism $\widetilde{E}\to E$ induced by $h$ is étale in codimension one; then the normality of $E$ implies that $\widetilde{E}$ is normal in codimension one. Thus $\widetilde{E}^{\prime}\to E$ is also étale in codimension one. Since $E$ is regular, the purity of the branch locus now implies that the finite morphism $\widetilde{E}^{\prime}\to E$ is étale at $\widetilde{x}^{\prime}$; but $\widetilde{x}^{\prime}$ is the unique point that lies above $x\in E$, so that $\widetilde{E}^{\prime}\to E$, and hence $\widetilde{E}\to E$, are isomorphisms. We finally conclude that $m=1$, so that $E$ is Cartier at $x$. ∎