The main result of the present paper is bounded elementary generation of the Steinberg groups \(\textrm{St}\hspace{0.55542pt}(\Phi ,R)\) for simply laced root systems \(\Phi \) of rank \(\geqslant 2\) and arbitrary Dedekind rings of arithmetic type. Also, we prove bounded generation of \(\textrm{St}\hspace{0.55542pt}(\Phi ,\mathbb {F}_{q}[\hspace{0.55542pt}t,t^{-1}])\) for all root systems \(\Phi \) , and bounded generation of \(\textrm{St}\hspace{0.55542pt}(\Phi ,\mathbb {F}_{q}[\hspace{0.55542pt}t])\) for all root systems \(\Phi \ne {\textsf{A}}_1\) . The proofs are based on a theorem on bounded elementary generation for the corresponding Chevalley groups, where we provide uniform bounds.
