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The purpose of a salt bridge is not to move electrons from the electrolyte, rather it's to maintain charge balance because the electrons are moving from one-half cell to the other. The electrons flow from the anode to the cathode.
Subsequently, one may also ask, why salt bridge is used in galvanic cell?
A salt bridge, in electrochemistry, is a laboratory device used to connect the oxidation and reduction half-cells of a galvanic cell (voltaic cell), a type of electrochemical cell. It maintains electrical neutrality within the internal circuit, preventing the cell from rapidly running its reaction to equilibrium.
Similarly one may ask, why dont electrolytic cells need a salt bridge?
In electrolytic cell you can dip both anode and cathode in the same solution. Then yes, you don't have a salt bridge because you use the same electrolyte. Salt bridge passes ions at a certain rate that is not too high in order to prevent electrolyte mixing. It has to pass ions at a right speed.
Without the salt bridge, the solution in the anode compartment would become positively charged and the solution in the cathode compartment would become negatively charged, because of the charge imbalance, the electrode reaction would quickly come to a halt.