Graphs without a rainbow path of length 3

In 1959 Erd\H{o}s and Gallai proved the asymptotically optimal bound for the maximum number of edges in graphs not containing a path of a fixed length. Here we study a rainbow version of their theorem, in which one considers $k \geq 1$ graphs on a common set of vertices not creating a path having edges from different graphs and asks for the maximum number of edges in each graph. We prove the asymptotically optimal bound in the case of a path on three edges and any $k \geq 1$.