Very difficult to diagnose over the net, but I'd start with the brake light lamp, on the same side that your turn signal and dash light is illuminating. Sounds like the lamp (bulb) is shorting out in the socket, or possibly internally, when you press the brakes, causing the back feed to illuminate the turn signal, and dash light. Doesn't sound like a ground issue, but again, tough to diagnose over the web.
This happened to me before, and I had a bad brake light lamp. It was a dual element, and shorting out internally, meaning one filament had failed, and was touching the other filament. Changed the lamp, and fixed the issue.
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