Steering column connector

Currents should be <5 A since your bulbs are maybe 50 W max (brake & turn filaments). The horn and starter wires trigger relays (~50 mA). You should always describe your car when asking such questions. I expect a 1970's w/ key in steering column. I recall that is a 1-row ~10 position connector from what I've seen in junkyards. My 1960's Mopars use a 2-row connector with "twin-lock" pins. Your bullet terminals look more common. Search for websites that sell automotive terminals. I don't have at hand. Try searching "56 terminal" (common type).
The connector and wires in question are ignition switch and not signal switch. That large black wire feeds the fuse box to high amp draws like wipers and A/C heater blower, plus radio, signals, etc...
For that one to overheat is somewhat common. Brown wire is ignition, hot in start. Blue is ignition hot in run plus charging system, choke heater, etc... So I have to wonder how much aftermarket equipment is involved here. MSD? Blue and brown tied together? Electric choke?