The graph invariant twin-width was recently introduced by Bonnet, Kim, Thomassé, and Watrigan. Problems expressible in first-order logic, which includes many prominent NP-hard problems, are tractable on graphs of bounded twin-width if a certificate for the twin-width bound is provided as an input. Computing such a certificate, however, is an intrinsic problem, for which no nontrivial algorithm is known. In this paper, we propose the first practical approach for computing the twin-width of graphs together with the corresponding certificate. We propose efficient SAT-encodings that rely on a characterization of twin-width based on elimination sequences. This allows us to determine the twin-width of many famous graphs with previously unknown twin-width. We utilize our encodings to identify the smallest graphs for a given twin-width bound $d \in \{1,\dots,4\}$.