The piecewise-smooth homotopy

Question

Let $M$ be a smooth manifold. If $\gamma_1:I \rightarrow M$ and $\gamma_2:I\rightarrow M$ are two piecewise-smooth homotopic curves (rel to endpoints), then can we find a map $f:I\times I\rightarrow M$ such that $f(0,t)=\gamma_1(t),f(1,t)=\gamma_2(t),f(s,0)=\gamma_1(0), f(s,1)=\gamma_1(1)$ and there is a finite partition $\{I_i\}$ of $I$ into subintervals so that $f$ is smooth on $I_i\times I_j$?