Choose a minimum feedback vertex set $S$. The remaining forest has a tree decomposition with at most two vertices in each bag: for a nonempty forest, extend it to a tree, root it, and use each vertex together with its parent as a bag, with a singleton bag at the root. For the empty forest, use one empty bag. Adding $S$ to every bag covers all original edges and preserves connectedness of the bags containing each vertex. Each new bag has at most $|S|+2$ vertices, so its width is at most $|S|+1$.