Mad Sci Guy, I'm afraid you have some critical facts wrong. The Hudson River and East River (or really the Harlem River, once you get north of Randalls/Wards Island) have always been connected above Manhattan Island. Manhattan was not a penninsula.
The original link was the Spuyten Duyvil Creek, a winding and shallow water passage, to be sure, but certainly *not* a landbridge! To prove the point, real bridges were built to cross it, like the famed Kings Bridge and the Farmers' (or Free) Bridge which were erected in colonial days and bore many a Revolutionary War troop movement.
As the city developed the Spuyten Duyvil proved a too-long and inadequate water link between the Hudson and Harlem, so the US Ship Canal (aka the Harlem River Ship Canal), which opened in 1895, was dug to make the connection shorter, deeper and more direct. (The canal sliced Marble Hill from Manhattan Island, and therein lies another tale, but not for this post!) In subsequent decades the canal was improved to its present state.