For a loose metric $d$ as above, we can consider the function
$$d_1(x,y):=\sup\{|d(x,z)-d(y,z)|;z\in X\}.$$

It is easy to verify that $d_1$ is a metric, and $d(x,y)\leq d_1(x,y)\leq\rho(d(x,y))$ for all $x,y$, so $d_1$ and $d$ generate the same topology.