In this case the current is flowing in forward direction due to which the force due to magnetic field acts towards the right and the rod moves towards right.
Here in this case, I say force causing motion is due to magnetic field and motion is too in its direction but all the time at all the places I have heard and read that "Work done by magnetic field is always zero" . If you say the statement is true then can you please help me in finding my solution.
read that "Work done by magnetic field is always zero"
This is misleading / not true in general. In macroscopic physics, (macroscopic) magnetic forces do work, for example magnetic force due to external field acting on current-carrying conductor $BIL$ does work when the conductor moves (such as in your example) and also this force can do work on magnetic bodies such as iron/steel bodies or magnets.
However, the quoted statement has a core of truth: work of magnetic force acting on a moving point charged particle (so-called Lorentz force) is always zero, because the force is always perpendicular to velocity.
This is consistent with the macroscopic physics, because work of macroscopic magnetic force on moving current-carrying conductor or magnetic body isn's simply sum of works of Lorentz forces due to external magnetic field; there are involved also induced internal forces between the current-forming charged particles and the rest of the conductor and these do the work. In current-carrying conductor, you can imagine them to be all the electric and non-electric forces that act between the mobile charge carriers (electrons) and the rest of the conductor.
