[dmsfrr]

Just to rule this out if nothing else...
Double check how the turn signal fixtures on that side are mounted to the car and connected to the wiring. There may or may not be a dedicated ground wire from the signal & parking/tail light housing to the car body. Loosen or clean and re-tighten the housing mounting screws to be sure they are making a good ground connection. Or, run some temporary alligator clip ground wires from the housing to the car body.

You'd think if the bulbs light up the ground connection must be good. 98% of the time you'd be right. But if a ground connection has high resistance / poor contact, various strange things can happen.

This (before) photo is of a '55, your '57 may or may not be similar. Notice the headlight has it's own ground wire but the other wires for the turn/parking light fixture do not. It electrically grounds thru its mounting screws. The tail light housings do also.