The construction of the Brooklyn Bridge was, in the end, made possible via a
bribe of around $60,000 payable to the
aldermen of New York City (meaning, at that time,
Manhattan). The money was provided by William C. Kingsley, the richest contractor in the City of
Brooklyn.
Additionally, Kingsley bought half of the Bridge Company's private stock and gifted it to William Magear "Boss" Tweed (who was the head of the Public Works department at the time) in order to get final approval. The inevitable value of the stock itself was garnished by the fact that Tweed could now hand out bridge work to his own people and let the kickbacks roll in.
Kingsley was gambling that these generous bribes would allay any fears Tweed or the aldermen might have about easy access to Brooklyn siphoning away wealthy home buyers from Tweed's Upper East Side real estate developments.
Information distilled from Gotham: A History of New York City to 1898 by Edwin G. Burrows and Mike Wallace (no, not that Mike Wallace.)