On the Turán number of the directed path

In this note we determine the maximum number of arcs in a digraph on $n$ vertices that does not contain a length-$k$ directed path ${\overrightarrow{P}}_{k+1}$ for $n\ge 50k^6$. This improves a theorem of Zhou and Li [$\mathit{\text{Graphs}}\mathit{\text{Combin.}}$ 39(3), 2023] who proved the result with a superexponential threshold on $n$.