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According to Google’s ngram viewer, a measure of word frequency in scanned books, “bedbugs” as a single word is used far more frequently:

ngram

Yet Google trends, which measures search terms, reports the converse:

trends

So authors are writing about bedbugs, but the general populace is searching for bed bugs.

Which is correct?

The convention among American entomologists is to treat common names where the insect category (“bug”) is correct as separate words, but to combine names into single words when the insect category is incorrect. Example: the house fly is really a fly, but a firefly is really a beetle. Since bed bugs belong to Hemiptera- the true bugs- then in our country, at least, the appropriate spelling is bed bug as two words.

Score one for the internet.