The electrons do not enter the light bulb filament with relative high kinetic energies.The potential energy of the electrons is predominantly converted by collisions with the filament crystal lattice into thermal energy and this is partly converted into light. Thus The mean speed of electrons doesn't change appreciably along the circuit path between the positive and negative terminal of the battery (and the filament in between). So, The current is constant along the path (no change between the 1st and 2nd half of the circuit).