The answer to why it's such a commonly misspelled word is likely due to the fact that it's been being misspelled ever since its inception as a word.
etymonline shows it originating in the mid-15th century, but there are books that have the misspelled version dating all the way from the 16th century, such as in this French book from 1581:
Legende de Domp Claude de Guyse, abbé de Cluny: contenant ses faits et ...
or from this Scottish book from 1585-1592, which has three instances of the misspelling, so you know it's not just a typo:
Scotland. Privy Council - 1585
Google Books shows the misspelling thriving from the 18th century-on:
An interesting thing that can be picked up from the sources is that in the mid-18th century the misspelling made its appearance in many French texts, and an Ngram analysis of French documents confirms this:
Google Ngrams French
Now is this a case of the British misspelling marking its influence on French soil, or was the misspelling French in origin all along? I doubt the latter, but maybe someone else can shed some light. French was the lingua franca during the 17th century so its possible.
There are various English words that derive from Latin words beginning with "exh-":
exhonorate is another common misspelling of exonerate, but that has a Latin root beginning with "exh", unlike "exorbitant" and "exonerate".
My guess after all this evidence is that the medieval Brits thought that "exorbitant's" Latin roots had an 'h' in them, which made it a common misspelling at the time, and that misspelling survived through the ages. Truly a misspelling for the ages.