From the Jewish Encylopedia on synagogues. Are they making stuff up?:
"The origin of the synagogue, in which the congregation gathered to worship and to receive the religious instruction connected therewith, is wrapped in obscurity. By the time it had become the central institution of Judaism (no period of the history of Israel is conceivable without it), it was already regarded as of ancient origin, dating back to the time of Moses (see Yer. Targ., Ex. xviii. 20 and I Chron. xvi. 39; Pesiḳ. 129b; Philo, "De Vita Mosis," iii. 27; Josephus, "Contra Ap." ii., § 17; Acts xv. 21). The "house of the people" (Jer. xxxix. 8 [Hebr.]) is interpreted, in a midrash cited by Rashi and Ḳimḥi (ad loc.), as referring to the synagogue, and "bet 'amma," the Aramaic form of this phrase, was the popular designation in the second century for the synagogue (Simeon b. Eleazar, in Shab. 32a). The synagogue as a permanent institution originated probably in the period of the Babylonian captivity, when a place for common worship and instruction had become necessary. The great prophet, in the second part of the Book of Isaiah, in applying the phrase "house of prayer" to the Temple to be built at Jerusalem (Isa. lvi. 7 and, according to the very defensible reading of the LXX., also lx. 7), may have used a phrase which, in the time of the Exile, designated the place of united worship; this interpretation is possible, furthermore, in such passages as Isa. lviii. 4. The term was preserved by the Hellenistic Jews as the name for the synagogue (προσευχή = οἶκος προσευχῆς; comp. also the allusion to the "proseucha" in Juvenal, "Satires," iii. 296).
After the return from the Captivity, when the religious life was reorganized, especially under Ezra and his successors, congregational worship, consisting in prayer and the reading of sections from the Bible, developed side by side with the revival of the cult of the Temple at Jerusalem, and thus led to the building of synagogues. The place of meeting was called "bet ha-keneset," since an assembly of the people for worship was termed a "keneset"; the assembly described in Neh. ix.-x. was known in tradition as the "great assembly" ("keneset ha-gedolah"; see Synagoġue, The Great). The synagoguecontinued to be known by this name, although it was called also, briefly, "keneset" (Aramaic, "kanishta"), and, in Greek, συναγωγή."
http://www.jewishencyclopedia.com/articles/14160-synagogue
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You are referring to Talmudic sources. They are not historical and not reliable.
This guy is an historian and reliable.
Solomon Zeitlin: “During the Second Commonwealth there were no synagogues as houses of prayer in Judea. The word synagogue does not occur in connection with prayers in the literature of that period.”