I've run Tower of the Elephant in B/X at low level. Tower of the Elephant is really just climbing, traps, seeing a weird god, and then finding the McGuffin so you can bring the whole thing down.This is true. If you wanted to run a conan-type adventure the minimum level you'd probably be looking at would 5th.
Conan adventures where he's killing 50 dudes is something you obviously can't do. But he has plenty of thief moments, and those thief moments are easy to replicate.