The process of fast magnetic reconnection supported by the formation of plasmoid chains in the high Lundquist number ($S$) regime is investigated using a recently developed adaptive finite-element magnetohydrodynamic (MHD) code. We employ a two-dimensional incompressible model with a set of reduced visco-resistive MHD equations. The tilt instability setup is chosen to provide a three-step mechanism, where two curved current sheets initially form on an Alfv\'enic time scale followed by a second phase of super-Alfv\'enic growth of plasmoid chains for $S \ge S_c$ (Baty 2020). A third phase is reached where an ensuing stochastic time-dependent reconnection regime with a fast time-averaged rate independent of $S$ is obtained. We reveal the multi-scale current structures during magnetic reconnection, where merging events of plasmoids give rise to monster plasmoids with shocks bounding the outflow regions. At high enough $S$ values (typically for $S \sim 100 S_c$), a dynamical Petschek-type reconnection is achieved with pairs of slow-mode shocks emanating from a small central region containing a few plasmoids. Finally, we briefly discuss the relevance of our results to explain the flaring activity in solar corona and internal disruptions in tokamaks.