'65 Dart No brake lights. Brake light switch failure?

No you guys are AFU. Here's how they work. The brake light switch gets "hot" power at all times, regardless of ignition switch. When you press the brake, the output (usually white) of the switch goes TO the turn signal switch. When centered, the TS switch routes this power through the switch to both rear lamps.

When turning, the TS switch disconnects one brake light (for the turn) and routes the TS flasher output to the front lamp (of course) and to the rear lamp, on the side meant to signal. The opposite brake lamp is routed from the brake light switch so you have a brake on the "non turn side.

THIS IS HOW ALL VEHICLES work with one set of lamps, unless it is some New fangled transistorized vehicle make recently that I'm not aware of.

Some cars (like my 70RR) with four rear lamps just paralleled them, IE two 1157 combo bulbs on each side, wired to work in unison

Other cars with SEPARATE brake lights (such as cars with amber turn signals in rear, and this started with foreign cars) do NOT use the TS switch connection. They simply run the brake switch straight back to the separate stop lamps, and the amber turn ones are run direct up to the TS switch.