Slightly generalizing, we may state reasonable conditions in terms of $G(x,y):=g(x,h(y))$ which implies boundedness of the sequence $(x_i)$, that is all we need. For instance:
If $G$ is continuous and $G(x,x)=0$ has a unique root $x^*$, then any sequence $(x_i)$ such that $G(x_{i+1}, x_i)=0$ does converge to $x^*$, provided $G$ also satisfies: there is $M$ such that $G(x,y)\neq0$ for any pair $(x,y)$ such that either $\min(x,y)> M$ or $\max(x,y)< -M$.
Indeed, in such a situation, one first observe that the sequence $x_i$ must be bounded (there can't be a subsequence $x_{i_j}$ converging to $\pm\infty$). So by compactness any subsequence of $x_i$ possesses a converging sub-sub-sequence. By continuity of $G$, the limit is a solution of $G(x,x)=0$, hence it's $x^*$, and this implies that the sequence $(x_i)$ itself converges to $x^*$.