redchevy is correct that lack of a good ground is a problem for a lot of trailers, but that isn't your problem (currently). Lack of good Ground may affect lights working, but it doesn't blow fuses. Besides, you say the brake/turn filaments illuminate correctly.

ChuChu1 is correct that you have a problem with the brown wire. SOMEWHERE along that wire, the insulation is damaged/cut, and there's a short to the trailer frame. There could be corrosion on a bulb base, but that's a lower priority. The brown wire is physically exposed to being hit by debris (even a tire self-destructing), so it has probably gotten damaged. Checking resistance from brown wire to Ground is tricky due to numerous parallel filaments in bulbs. You would really need to remove ALL of the bulbs, including marker lights. It's easier to just visually trace the wire to look for the problem.

One other slim possibility: Have you replaced a bulb? If you have, remove that bulb and try the lights again to see if they work and if the fuse still blows. An incorrect bulb can short circuits that shouldn't be, as can an improperly installed bulb. I've seen bulbs incorrectly installed that blew fuses instantly when lighting was turned on.