Let $X$ be a connected normal scheme equipped with a proper flat morphism $f\colon X \rightarrow \mathrm{Spec }(R)$ with $R$ a discrete valuation ring and such that the fibers of $f$ are curves (i.e., purely of dimension $1$). Is $f$ necessarily projective?
The answer is 'yes' if the normality assumption is strengthened to regularity of $X$: this was proved in Lichtenbaum's thesis.
I believe that you can form a counterexample in the same manner "as usual". Begin with a projective morphism $\pi:\mathcal{C}_R\to \text{Spec}(R)$ whose closed fiber $\mathcal{C}_k$ over the residue field $k$ is a smooth, geometrically connected curve of genus $g>0$. For simplicity, assume that $\text{Pic}(\mathcal{C}_R)$ is $\mathbb{Z}$ and assume that $\mathcal{C}_k$ has a $k$-point that is linearly independent from the image of $\text{Pic}(\mathcal{C}_R)$ in $\text{Pic}(\mathcal{C}_k)$.
For instance, let $d\geq 3$ be an integer, let $\mathbb{P}^N = \mathbb{P}H^0(\mathbb{P}^2,\mathcal{O}(d))$ be the projective space of all degree $d$ plane curves, let $D_q\subset \mathbb{P}^N$ be the hyperplane parameterizing degree $d$ plane curves that contain a specified point $q\in \mathbb{P}^2$, let $\mathcal{C} \to \mathbb{P}^N$ be the universal curve, and let $R$ be the stalk $\mathcal{O}_{\mathbb{P}^N,\eta_H}$ at the generic point $\eta_H$ of $H_q$. Let $\mathcal{C}_R$ be the base chance of $\mathcal{C}$ to $\text{Spec}(R)$. For the fraction field $K$ of $R$, $\mathcal{C}_K$ has Picard group generated by the restriction of $\mathcal{O}_{\mathbb{P}^2}(1)$, by a straightforward incidence correspondence argument. Yet, by construction, $\mathcal{C}_k$ contains the $k$-point $q$. Moreover, the divisor class of $\underline{q}$ is linearly independent from the restriction of $c_1(\mathcal{O}_{\mathbb{P}^2}(1))$ in $\text{Pic}(\mathcal{C}_k)$.
Anyway, now let
$$
\nu:\mathcal{C}'_R \to \mathcal{C}_R
$$
be the blowing up at the closed point $q$ (with its reduced structure).
The strict transform $\widetilde{\mathcal{C}}_k$ has normal sheaf $\mathcal{O}_{\mathcal{C}_k}(-\underline{q})$. By Grauert / Artin, there exists a contraction
$$
\mu:\mathcal{C}'_R \to \mathcal{C}''_R
$$
of $\widetilde{\mathcal{C}}_k$ in $\mathcal{C}'_R$. I claim that the projection morphism to $\text{Spec}(R)$ factors through $\mu$,
$$
\pi'':\mathcal{C}''_R \to \text{Spec}(R).
$$
If $\mathcal{C}''_R$ were projective, the pullback to $\mathcal{C}'_R$ of an ample invertible sheaf would restrict on $\mathcal{C}_K$ to an integer multiple of the generator of $\text{Pic}(\mathcal{C}_K)$. Yet on $\widetilde{\mathcal{C}}_k$ it restricts to the structure sheaf. Since the exceptional divisor on $\mathcal{C}'_R$ restricts on $\widetilde{\mathcal{C}}_k$ to $\mathcal{O}_{\mathcal{C}_k}(-\underline{q})$, ultimately this implies a linear relation between $\text{Pic}(\mathcal{C}_R)$ and the divisor class of $\underline{q}$. This contradiction implies that $\mathcal{C}''_R$ is not projective.
Edit. As Laurent Moret-Bailly points out, the surface $\mathcal{C}''$ produced above is usually an algebraic space, not a scheme.
This is perhaps not the answer the poster is looking for, but Proposition 3.3 of Families of rationally simply connected varieties over surfaces and torsors for semisimple groups, by de Jong--He--Starr, proves something related. Indeed, an immediate consequence of this result is that if $\pi \colon C \to S$ is a proper, flat, and finitely presented morphism of algebraic spaces of relative dimension 1, then there exists an etale cover $S' \to S$ such that the pullback $\pi' \colon C\times_S S' \to S'$ is projective.
In particular, in the notation of the posed question: if $R$ is strictly henselian (e.g., complete with algebraically closed residue field), then $f$ is projective.
Note that it is not even necessary to assume that $X$ is normal for this this result to hold.
