  The main result of the present paper is bounded elementary generation of the
Steinberg groups $\mathrm{St}(\Phi,R)$ for simply laced root systems $\Phi$ of
rank $\ge 2$ and arbitrary Dedekind rings of arithmetic type. Also, we prove
bounded generation of $\mathrm{St}(\Phi,\mathbb F_{q}[t,\,t^{-1}])$ for all
root systems $\Phi$, and bounded generation of $\mathrm{St}(\Phi,\mathbb
F_{q}[t])$ for all root systems $\Phi\neq\mathsf A_1$.
  The proofs are based on a theorem on bounded elementary generation for the
corresponding Chevalley groups, where we provide uniform bounds.
