A search for heavy charged Higgs bosons decaying to a pair of top and bottom quarks is presented. The analysis is performed using 139 $\text{fb}^{-1}$ proton-proton collision data collected with the ATLAS detector at the Large Hadron Collider. The production of a heavy charged Higgs boson in association with a top and a bottom quarks, $pp\rightarrow tbH^+\rightarrow tb tb$, is explored in the $H^+$ mass range from 200 to 2000 GeV using multi-jet final states with one electron or muon. A mass-parameterised neural network is used to discriminate between signal and background events in four separate regions categorised according to the multiplicity of jets and jets originated from hadronisation of a bottom quark. No significant excess above the background-only hypothesis is observed and upper limits on the production cross-section times branching ratio of the charged Higgs boson as a function of its mass are derived. The results are interpreted in the context of the hMSSM and $\text{M}_{\text{h}}^{125}$ scenarios of the Minimal Supersymmetric Standard Model.