Mine did the same thing. My mower would not start at all when I turned the key so I took a screwdriver and touched both negative and positive connections on the solinoid and it turned over and started. I figured my key switch was out. I did check the fuse and it was fine. That's how I got my briggs & stratton 21 horse twin-V started. After some work it runs fine now.

First off, you didn't cross the positive and negative, they are both positive. Positive from the battery to the solenoid, and positive from the solenoid to the starter when the key switch is activated.
Your solenoid is probably bad.