I know these problems can be frustrating, especially if you don't have the tools and good electrical background to fall back on. No problem though - the folks on this forum are willing to help.
Let's back up a bit and start over. I think we approached this with the assumption that you had an electrical background. I apologize.
With respect my friend, you've really been describing symptoms that are most common with a miss-wire situation (It really sounds like you mixed up the yellow and brown wires on the driver's side). That's why
@Babin Farms sent you the standardized color code and I suggested you actually test the lights in case the trailer had some unusual wiring. The light you sent back was likely still good. Ground issues are indeed a common nuisance and a Go-To fault, but with incandescent bulbs, grounding issues generally cause an intermittent or failure to light. With LED's it gets a little trickier sometimes due to polarity issues, but they aren't likely to cause the symptoms you described. All is good because you can troubleshoot all of these issues I mentioned with an inexpensive automotive test light from any box or automotive store.
https://www.amazon.com/WINAMOO-Automotive-Extended-Stainless-Circuits/dp/B08NWQWQBZ/ref=sr_1_9?dchild=1&keywords=automotive+test+light&qid=1616714749&sr=8-9
@Babin Farms gave you the standardized wiring color code. Print it and take it to the trailer with you. Test the 12v Automotive Tester the first time by putting the clamp end on the negative terminal of your battery and the sharp end on the positive terminal. The light should light fairly brightly. That's a closed and correct circuit. In essence, that's all your doing at the end of the tail light wires as well. Simply connect the ground clip of the test light or the negative lead of your voltmeter to a clean/bare metal area of your trailer chassis and activate the light circuit you want to test (Turn signal, brake, etc) from inside the vehicle, then poke the exposed trailer wire with the sharp end to check to see if you get the voltage on the correct wire as indicated by a light or a 12vdc reading on the meter. You'll know its working if it glows or blinks according to the light circuit you're testing. Do one light circuit at a time until you know which wires do what. Side Note: Also watch for a dim light indication. If you get that on clean wires with good contact, shout back out on this thread - we can help you fix that too. If you don't get the proper voltages or they are intermittent, that suggests that you have a broken wire or a ground issue which can be verified easily by going back to validate it's coming out of the vehicle correctly. If you ever confirm a ground issue on the trailer (i.e. the vehicle lights properly, but your trailer wires don't), you can simply chase down where the white wire connects to the trailer (Probably with a lug under a big bold near the tongue). Look for broken strands in your wire or corrosion which can be fixed by pulling the bolt, wire brushing and reinstallation, perhaps with some dielectric grease if you want.
If your tests on the tail light wires work out right away and you validate that your trailer is wired with the standardized color code
@Babin Farms gave you, then just connect the wires to the light per the manufacturer's instructions.
If I failed to explain it properly, you might want to check out YouTube. I'm pretty sure that there will be a video that can show it much clearer. A photo really is worth a thousand words. One other solution to consider: If this isn't in your comfort zone, for safety reasons it may be best to just to take it to a local trailer company or even a quickie repair shop like Car-X, Meineke, or Midas. Most shops will even let you watch from a safe distance so you can learn with an live instructor. I've learned a lot from the professionals that way. You certainly don't want to be on the road with a fix that didn't hold. Good Luck my friend.
