2002 Ford explorer won't start battery good new starter

Sounds like the ignition switch is bad.

or solenoid /relay

Get yourself a test light or vlot meter,have one turn key to "start" while 2nd person checks for power where you did your screw driver trick. That should narrow you problem down.

The starter/relay which may be bolted on the firewall on the same side of the battery is probably shot. You can jump that also, but if it starts, let it go immediately or you man damage your starter.