Doing a search of the site, there are these articles:"2000 verses"
and "two thousand verses"
As far as an official history on this topic, "God Passes By", written by Shoghi Effendi, has the following:
The enormous expansion in the scope and volume of Bahá'u'lláh's writings, after His return from Sulamáníyyih, is yet another distinguishing feature of the period under review. The verses that streamed during those years from His pen, described as "a copious rain" by Himself, whether in the form of epistles, exhortations, commentaries, apologies, dissertations, prophecies, prayers, odes or specific Tablets, contributed, to a marked degree, to the reformation and progressive unfoldment of the Bábí community, to the broadening of its outlook, to the expansion of its activities and to the enlightenment of the minds of its members. So prolific was this period, that during the first two years after His return from His retirement, according to the testimony of Nabíl, who was at that time living in Baghdád, the unrecorded verses that streamed from His lips averaged, in a single day and night, the equivalent of the Qur'án! As to those verses which He either dictated or wrote Himself, their number was no less remarkable than either the wealth of material they contained, or the diversity of subjects to which they referred. A vast, and indeed the greater, proportion of these writings were, alas, lost irretrievably to posterity. No less an authority than Mírzá Áqá Ján, Bahá'u'lláh's amanuensis, affirms, as reported by Nabíl, that by the express order of Bahá'u'lláh, hundreds of thousands of verses, mostly written by His own hand, were obliterated and cast into the river. "Finding me reluctant to execute His orders," Mírzá Áqá Ján has related to Nabíl, "Bahá'u'lláh would reassure me saying: `None is to be found at this time worthy to hear these melodies.' ...Not once, or twice, but innumerable times, was I commanded to repeat this act." A certain Muhammad Karím, a native of Shíráz, who had been a witness to the rapidity and the manner in which the Báb had penned the verses with which He was inspired, has left the following testimony to posterity, after attaining, during those days, the presence of Bahá'u'lláh, and beholding with his own eyes what he himself had considered to be the only proof of the mission of the Promised One: "I bear witness that the verses revealed by Bahá'u'lláh were superior, in the rapidity with which they were penned, in the ease with which they flowed, in their lucidity, their profundity and sweetness to those which I, myself saw pour from the pen of the Báb when in His presence. Had Bahá'u'lláh no other claim to greatness, this were sufficient, in the eyes of the world and its people, that He produced such verses as have streamed this day from His pen."
Also on p. 123 of the same book
, another episode of Baha'u'llah penning 2000 verses is described (but only mentioning the verses being destroyed):
Amazed by the profundity of His insight and the compass of His understanding, they were impelled to seek from Him what they considered to be a conclusive and final evidence of the unique power and knowledge which He now appeared in their eyes to possess. "No one among the mystics, the wise, and the learned," they claimed, while requesting this further favor from Him, "has hitherto proved himself capable of writing a poem in a rhyme and meter identical with that of the longer of the two odes, entitled Qásidiy-i-Ta'íyyih composed by Ibn-i-Faríd. We beg you to write for us a poem in that same meter and rhyme." This request was complied with, and no less than two thousand verses, in exactly the manner they had specified, were dictated by Him, out of which He selected one hundred and twenty-seven, which He permitted them to keep, deeming the subject matter of the rest premature and unsuitable to the needs of the times. It is these same one hundred and twenty-seven verses that constitute the Qásidiy-i-Varqá'íyyih, so familiar to, and widely circulated amongst, His Arabic speaking followers.
Such was their reaction to this marvelous demonstration of the sagacity and genius of Bahá'u'lláh that they unanimously acknowledged every single verse of that poem to be endowed with a force, beauty and power far surpassing anything contained in either the major or minor odes composed by that celebrated poet.
Not sure if the above might have been accidentally merged in the minds of those writing about the 2000 verses and the river together or whether in the latter example, they were cast into the river...