2003 Chevy Silverado When I apply the brakes the

First you need to have EBCM - electronic brake control module check for DTC'S - diagnostic trouble code's . If the light is on there is a code stored . I think most part stores can do this and for free . After you find out what code is stored , looking up that code in service repair information for diagnostic trouble shooting chart would be next step . There are a few codes too.
DTC C0035 or C0040
DTC C0045 or C0050
DTC C0110
DTC C0131
DTC C0161
DTC C0186
DTC C0196
DTC C0201
DTC C0240
DTC C0241 or P0856
DTC C0244 or P1689
DTC C0245
DTC C0283
DTC C0287
DTC C0290 or C0292
DTC C0455
DTC C0550
DTC C0558

Find the code do the diagnostic's , fix the problem , wiring issue's , a sensor , bad module ,what ever it is . The light will go out then . Here is a sample of code discription .
• DTC C0035 Left Front Wheel Speed Sensor Circuit
• DTC C0040 Right Front Wheel Speed Sensor Circuit

sounds like you are getting water into the connections and possibly have a bad earth connection. Suggest you check all light connections for water ingression.

It could be the turn signal switch. the white wire comes from the brake switch into the turn signal switch. Make sure that is hot with brakes applied. the wires coming out of the turn signal switch to the left is Yellow, and Dark Green for the right. Same for the turn signals.

More than likely a bad ground in the wiring harness's. Have you changed your transmission, Alternator, or starter? Or added a trailer plug on the bumper?

Power for left rear turn signal is dark blue
Power for left rear tail light is brown
Power for left rear stop light is light blue.
Black is ground

There are 2 filiments in the light bulb. One is for running lights and the other is for brake and turn signal lights. Simply replace the bulb and you should be ok.

Sounds like the turn signal flasher module has gone bad. It controls the brake lights and turn signals. It is located under the drivers side of the dash. If is a rectangular black box with a grey connector and 8 or so wires connected to it.

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97 chevy blazer left turn signal makes the dash lights flash then when i turn parking lights on left signal stays on constent when engaged

Check under hood fuse box make sure L. trailer light fuse isnt blown.

if you have the circuit board type rear lights the circuits inside will brake and cross eachother that will cause your problem