Interesting. What's your reasoning for thinking this? I can buy that fascism is extremist conservatism, but I'm struggling to see how fascism is the natural conclusion of conservatism.
I should probably have been a little clearer but I wanted to be a wiseass and make a short, snappy comment. I don't necessarily think that all conservatives are Nazis and I think fascism is generally a sort of spectrum, and most (if not all) people have some fascistic tendencies that they need to be aware of and try to compensate for (I certainly do).
Here's an interesting video that touches on a similar concept (potential fascism as opposed to actual fascism). In my opinion, conservative ideologies usually find themselves much deeper on that spectrum.
That being said, I do think that fascism is the logical endpoint of conservatism most of the time (the only times it isn't is when that particular conservative ideology is held in check by non-conservative values). As bobjr said, every variant of conservatism relies either on an underclass or an out-group to function, and radical ideas of racial, ethnic, and ideological purity are the natural endpoint of such thinking.
In a way, I feel like you kind of admit this yourself. You say that you can accept that fascism is extremist conservatism. Doesn't that make it the natural conclusion of conservatism? If you allow most ideologies to play out unchecked by outside influences, do they not end up at their most extreme variants? I think it's kind of baked into human nature that if you think something is right, you have no incentive to stop doing it.
This is why diversity is necessary for a healthy society to function as it does introduce these checks and restraints that, in theory, help prevent extremist ideologies from emerging. In fact, I think it's telling that, historically, the first step for fascist parties has been to attempt to undermine and invalidate the very concept of diversity. The Nazis sold the idea of racial purity, which is fundamentally incompatible with diversity and offered a platform for their most abhorrent ideas about the inferiority of other races and ethnicities. They eroded what was keeping the conservative ways of thinking that many German citizens already had from reaching its logical endpoint, and so it did, and millions of innocent people paid the price for it.
Conservatism is always adversarial to diversity (as we've said, it relies on an out-group or underclass of some kind to function), so I don't think it's much of a logical leap to say that fascism is the natural endpoint of conservative ideologies.