It seems unavoidable. Interestingly (and in my totally unscientific opinion), I believe it's one of the earliest signs that a convert has really affiliated with the Jewish people.
My theory can be analogized to linguistics. You know you really know a language when you begin to dream in that language. Personally, I've progressed that far with Spanish and French (though I can't vouch for being anywhere near fluent in either anymore). It's a really strange feeling. Even within the dream, I'd stop and think it was strange that I was speaking and listening solely in a foreign language. When I woke up, I knew exactly what had happened in the dream, even if I couldn't reproduce the "script," so to speak.
This is called internalization. You have internalized the language such that it has become a part of you. However, you don't have to be fluent to reach that stage. But you won't reach that stage if you insist on thinking in English, then translating the English thought into the target language. Internalization is when you begin to actually think in that language like a native speaker would. [Personally, I think this is the secret to learning languages, and anyone can force themselves to do it!] It takes a lot of humility to admit to yourself that you can really only say, "I like cheese" instead of discussing the amazing cheese dish you just cooked.
So what does that mean for converts/conversion candidates? When you "think Jewish," you will begin to internalize a Jewish identity. That is all kinds of awesome, but Jewishness also includes a history of antisemitism. The good comes with the bad. Thankfully, the only antisemitism many of us experience is in the dream world. That is certainly true for antisemitic violence, which is thankfully not common in America today. But that isn't very comforting while Nazis hunt you down in the Polish countryside of dreamland! When you have that first dream, you will wake up thinking you're insane. But don't worry, every Jew has these dreams at least occasionally, according to my very unscientific polls.
This discussion doesn't make the nightmares any less disturbing, but maybe now you can see the good in them too. And if they freak you out or are otherwise bothering you, feel free to discuss them with your rabbi. I think that's an important topic to work through with him. Anti-Semitism is something you really need to come to terms with, and this is the way most of us do.