97' OLD'S CUTLASS SUPREME (TURN SIGNALS) Front and Rear LEFT & RIGHT TURN SIGNAL ONLY, Will not flash. When using the turn signal switch the lite stays on, I have to use it manually. ** 4-Way flashers works o.k. ** Replaced flasher under dash. ** Checked fuses and there o.k. ThanX - Tonester

The BCM - body control module controls turn signals . There is no flasher unit . Your best bet would be to take your vehicle to the dealer !
Turn Signal Lamps
The BCM 3, BCM 5, and BCM 6 fuses located in the underhood fuse block supply battery voltage to the body control module (BCM) for turn signal, hazard lamp, and stop lamp operation. Voltage from the BCM 5 fuse used for the front and rear left turn signals, voltage from the BCM 3 for the right front turn signal, while voltage from the BCM 6 fuse is used for the right rear turn signal. When the turn signal switch is place in either the LEFT or RIGHT position, ground from G303 is applied through the turn signal switch signal circuit to the BCM indicating the turn signal request. In response to this signal, the BCM applies a pulsating voltage to the front and rear turn signal lamps supply voltage circuits cycling the lamps ON and OFF. The BCM also sends a message via GMLAN to the instrument panel cluster (IPC) to cycle the turn signal indicator ON and OFF depending on the position of the turn signal switch.

Ignition ON, observe the scan tool Left Turn Signal Switch parameter while using the turn signal/multifunction switch to turn the left turn signal ON and OFF. The reading should change between Active and Inactive.

In the diagram for rear lamp, the top wires are voltage, the bottom black wire is ground. The brown wire on either side should be tail lamps, goes hot when you turn on the headlamp switch. The other top wire on either should be for turn signal. With everything turned on, is there voltage on that wire?
Because the front works and the rear don't, possible wiring circuit problem? The voltage for turn signals comes from bcm. The bcm knows when you turn on turn signal switch, either side, then the bcm sends voltage to turn signal lamp at the rear and front.

my oldsmobile 1995 cutlass supreme has the same problem, so I replaced turn signal switch.

Turn signal switch may be bad from what it sounds like.

most likely its your muti-funtion switch in the steering column.but first check your turn signal flasher & your emergency flasher

One of 2 things to have checked, if all your bulbs and connections are correct. 1-stop lamp switch module.
2-multifunction/combination switch (blinker arm/components that are in the steering column). For about a 1/2 hrs. labor cost, a mechanic should be able to test the circuits.

you need to replace the flasher unit

Disconnect the battery to do any service on the steering column or front radiator areas.

replace the switch