A search for charged Higgs bosons heavier than the top quark and decaying via $H^+ \to tb$ is presented. The data analysed corresponds to 36.1 fb$^{-1}$ of $pp$ collisions at $\sqrt{s} = 13$ TeV and was recorded with the ATLAS detector at the LHC in 2015 and 2016. The production of a charged Higgs boson in association with a top quark and a bottom quark, $pp \to tb H^+$, is explored in the mass range from $m_H^+ =$ 200 to 2000 GeV using multi-jet final states with one or two electrons or muons. Events are categorised according to the multiplicity of jets and how likely these are to have originated from hadronisation of a bottom quark. Multivariate techniques are used to discriminate between signal and background events. No significant excess above the background-only hypothesis is observed and exclusion limits are derived for the production cross-section times branching fraction of a charged Higgs boson as a function of its mass, which range from 2.9 pb at $m_H^+ =$ 200 GeV to 0.070 pb at $m_H^+ =$ 2000 GeV. The results are interpreted in two benchmark scenarios of the Minimal Supersymmetric Standard Model.

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