You may have already figured this out but.
When you use a screwdriver between the battery terminal and the Sol terminal (red) you are bypassing the starter relay and not testing the starter relay for functionality.
When you jumper from the batt terminal to the I (Ignition, IGN 2 whatever you want to call it) terminal (Green) you are bypassing the ignition switch but testing the starter relay for functionality.
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As to why it will start when the turn signal is on, I would assume you or someone has removed wires from the steering column connectors and put them back incorrectly.