While performing the controlled impedance routing what about the current flowing through it?

While calculating the conductor impedance the parameters that we consider are the conductor width, conductor thickness, frequency & substrate parameters. For achieving the particular impedance based on the material properties we are fixing the trace width. What about the current which depends on trace width ?

Luckily, the characteristic impedance depends mainly on the ratio $w/h$, where w is the width of the trace and h is the height of the trace above the ground plane. So you can make your trace wider and keep your characteristic impedance constant by increasing h. (Taking care not to increase the dimensions so much that your transmission line becomes multi-modal at your operating frequency)