2004 4runner, right rear turn signal does not work...

You simply have a burned out bulb or bulbs. With the key turned on turn on the directional for the left side and then the right side. Typically one bulb is blown and the opposite side works normally. Observe the normal side and note which is the flashing bulb. It is the flashing bulb on the other (malfunctioning) side that needs to be replaced. To replace the bulb you must remove the entire brakelight assembly, which isn't difficult. Open the trunk and locate the electrical harness that lines up with the brakelight assembly. Near the harness, locate the two (possibly three) nuts or wingnuts that attach the brakelight assembly to the car. Once the nuts are removed, the assembly can be pulled off the rear of the car. After replacing the bulb, be sure to test the operation of the new bulb before reassembling everthing.

Has the vehicle been in an accident on the left rear side. First check the bulb to be sure it is not defective, trade it out with the side that works properly to test. If the bulb works well then My thought are you may have a shorted wire. The turn signal flasher works by heat build up within the flasher the heat is caused by the load of the bulbs, the load heats the metal spring causing it to open the circuit then as quickly as it cools down the circuit closes again lighting the bulbs.
There is also a possibility of a defect in the turn signal mechanism on the steering column. The turn signal switch on many models does three different things on many models that use the same bulb element / or bulb for brake and turn on rear of vehicle. It controls the circuit for brake light by sending power both left and right side rear when turn signal is in neutral position then when turn signal is in left turn the switch cancels brake light power to left side rear and supplies flasher power to left side while maintaining brake light supply to the right side rear, and so on opposite when the right turn is chosen on turn signal switch. The turn signal indicators on the dash normally run off the same circuit as front turn signal bulb only which is also controlled by turn signal switch. Front has turn signals but no brake light.
I think there is an additional load on the left side rear turn circuit and it could be either a wire that is shorted or the turn signal switch if the same bulb element is used for both functions.
Often times if trailer lighting is connected the flasher will also blink a little faster because of the extra load of additional bulbs.

It seems to be a ground defect on right side. Do you have the same bulb's number . What about the front flasher. remove right rear lamp assembly remove connector measure voltage between ground and current .reconnect it measure voltage at bulb socket. you should find your problems
hope it helps

Must be a bad socket, anyway. If I understand, you are talking about the two contact points inside the socket? On a two filament bulb, one contact is for the smaller filament park light, and the other contact is for the turn signal and/or brake light, the larger filament inside the bulb. So if you tested with a test light, with park lights on, one contact only would light up the tester. If you pressed the brakes, the other contact would have a steady light on the tester. That same contact would or should have a blinking test light, with the key on and the right turn signal on.
I would not be surprised if you need a new assembly. Possibly, too, the ground for that side needs checked.

check the bulb there some bulbs have two elements in them the turn signal element might be broken change the bulb

If the indicators on the dash are staying lit all the time, it is because you likely have a bad ground to one of your running lights, causing the running lights to ground through the turn signal circuits. This is often caused by either a bad socket assembly at one of the bulbs or an incorrect bulb is installed in one of the sockets. You need to check FRONT and REAR lights to see which one is not working correctly. Also check the running lights and marker lights on BOTH sides of the vehicle. There will be SOMETHING that will not look the same from one side to the other if the left side is working correctly and the right side is not. Please keep in mind that you cannot use the right REAR turn signal without also using the right FRONT turn signal. The problem could be at EITHER end.

Does any lights on the right side work? It sounds like the ground connection to the assembly is loose.

You have a bad ground wire on the right rear.
if you need further info let me know.

First,

1. Turn the engine off and open the hood. Pry up the two fasteners from the radiator air baffle with a flat head screwdriver. Move the baffle to the side to access the top of the headlight case.
   
2. Step 2 Pull up the two metal rods behind the headlight case. These secure the case to the car. You may need your pliers to pull them up.
   
3. Step 3 Gently pull the headlight case out to access its rear. On the rear, you will find several black dust covers. Twist off the dust cover closest to the fender (the outward dust cover) to access the turn signal bulb.
   
4. Step 4 Turn the bulb module counterclockwise to release it. Unplug the electrical connector from the bulb module. Connect the new bulb to the electrical connector (don't touch the glass with your hands; hold it by the plastic piece). Insert the bulb module in the turn signal housing, turn it clockwise and screw the dust cover back on.
   
5. Step 5 Insert the headlight case in the mount. Push the pins back down to secure it. Position the radiator baffle in its place and insert the fasteners.
   
Rear Turn Signal Bulbs
6. Step 1 Open the trunk. Remove the net from the side mounted wing nut (on the side with the faulty turn signal). Twist the wing nuts (there are two per side) counterclockwise to remove them.
   
7. Step 2 Peel the carpet layer back to access the rear signal lamp wing nuts. Twist these wing nuts (two per side) counterclockwise to remove them and release the rear signal.
   
8. Step 3 Carefully pull the rear signal assembly away from the Impala. The rear turn signal is the inner bulb (closest to the trunk). Press the tab on the bulb module and turn it counterclockwise to release it from the rear signal assembly.
   
9. Step 4 Pull the old bulb straight off the module (don't twist). Insert the new bulb in the module. Depress the tab on the connection and insert the module back in the assembly. Twist it clockwise.
   
10. Step 5 Insert the rear signal assembly. Re-attach the assembly's wing nuts and push the carpet layer back in place. Tighten the other two wing nuts to secure the carpet layer and reconnect the cargo net
    

you replaced the bulb with the wrong one i.e. a two filament bulb with a one filament bulb