I don't remember precisely where I read about their views on "fantasy" stories of the time... it may have been in one of the Daily Tolkien threads, posted a long time ago. Here's a link to all of the threads; a number of the articles deal with Numenor, and some comment on Tolkien's idea of "sub-creation".
BTW, the reason I put "fantasy" in quotes above is because JRRT and CSL considered the newly-minted genre of Science Fiction to be fantasy stories; tales of imaginary worlds and all that. Back then, there wasn't the sharp distinction between SF and fantasy that we see today.
I've never read any of the Conan stories, but I can see where there might be parallels to JRRT's Middle-Earth, as IIRC both series in their original concept were set in pre-historical Northern Europe.
I agree there was no sharp fantasy/SF distinction at that time--I think SF itself as a distinct genre didn't emerge until the late 1920s/early 1930s, and even then wasn't sharply distinct from fantasy until fantasy emerged into its own right with the post-60s popularity of LotR.
The first part of Howard's essay on the historical background to the Conan stories is online here:
My theory is that Tolkien may have read this and been influenced by it. Jen, I'm pinging you on this because I know you haven't read Conan yet and I thought you might also find this interesting. BTW, there's a map that goes with this essay, but unfortunately it's not on this page--I'll see if I can find a copy online. See this page, though, for a good supplementary guide to Howard's essay: