There is an old pronunciation of "syndrome" with three syllables (stress on the first). John Walker's Critical Pronouncing Dictionary of 1791 only gives a trisyllabic pronunciation, "si²n´dro¹-me¹" (where i² = the vowel of "pin", o¹ = the vowel of "no", and e¹ = the vowel of "me"). (This corresponds to the transcription that Christopher Ford mentioned in a comment, /ˈsɪndrəʊmi/, given by the Longman Pronunciation Dictionary).
This pronunciation exists because syndrome can be viewed as a direct transliteration of Greek συνδρομή, with -e standing for a pronounced final vowel sound: compare syncope = συγκοπή, epitome = ἐπιτομή or Nike = Νίκη.
The words hippodrome and palindrome, in contrast, are not direct transliterations of the corresponding Greek forms: they originate from Greek words ending in -δρομος (ἱππόδρομος, παλίνδρομος 'hippodromos, palindromos').
I'm not sure exactly how and when the disyllabic pronunciation of syndrome arose. It seems likely that many people first encounter these words in writing, and so pronounce them as they are spelled according to the analogies of the English language. Aside from spelling pronunciation, it's possible the two-syllable pronunciation of syndrome was influenced by other languages such as French (where syndrome is always two syllables) or German (where the word has the form Syndrom).