A salt bridge is essential in the continuous operation of a galvanic cell.
As oxidation proceeds, the electrons produced are immediately pulled to the reduction site, leaving the oxidation site with a net positive charge caused by the more positive ions produced. On the other hand, in the reduction site, the more positive ions are combined with electrons, leading to a more negative charge in the reduction site.
A salt bridge (filled with dissociated electrolytes) can fix the charge buildup on both sides and return them to net neutral charge by sending some anions to the oxidation site and some cations to the reduction site. Without this balancing of charge, the reaction will fail to proceed, since any electrons produced by the oxidation will be attracted by the positive charge in the oxidation site and repulsed by the negative charge on the cathode’s side and cannot go over, which of course means no current produced.