A search for charged Higgs bosons heavier than the top quark and decaying via $H^{\pm} \to tb$ is presented. The analysis uses 13.2 $\mathrm{fb^{-1}}$ of $pp$ collision data at a centre-of-mass energy $\sqrt{s} = 13$ TeV collected with the ATLAS detector at the Large Hadron Collider. The production of a charged Higgs boson in association with a top quark and a bottom quark, $pp \to tbH^{\pm}$, is explored in the mass range 300 to 1000 GeV using multi-jet final states with one electron or muon from the top-quark decay. Events are categorised according to the multiplicity of jet and $b$-tagged jets. Multivariate techniques are used to discriminate between signal and background events. No significant excess above the background-only hypothesis is observed and upper limits are set on the $tbH^{\pm}$ production cross section times the branching fraction BR($H^ {\pm}\to tb$), together with their interpretation in benchmark scenarios of the Minimal Supersymmetric Standard Model.

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