The video shows one of the buses pulling into an intersection in Queens when it was struck.
The tour bus driver and a passenger on the transit bus were killed, as was a pedestrian.
Mayor Bill de Blasio said he was shocked by the "sheer destruction," and noted that "one of those who was lost was simply walking down the sidewalk" as the crash came "out of nowhere."
Sixteen other people were hurt, some of them seriously, in the crash at 6:15 a.m. in the Flushing, Queens neighborhood, city officials said at a briefing.
A building that houses a number of retail stores, including a fried chicken restaurant, also sustained a "very substantial" impact, and experts were working to make sure it was secure, de Blasio said.
The hospitalized city bus driver was being interviewed by investigators, Metropolitan Transportation Authority Chairman Joe Lhota said.
Though the investigation has just begun, "These buses spun around. That requires an enormous amount of speed," Lhota said.
"We want to make sure we understand what happened and prevent this from ever happening again," Lhota said.
Signage on the tour bus showed it was from the Dahlia Group, Inc., based in Flushing. A person answering the phone there declined to comment; there was no immediately response to an emailed comment request.
Federal Motor Carrier Safety Administration records show that a Dahlia bus was also involved in a fatal crash in Connecticut in February 2016.
One of the company's buses was driving through a snowstorm to reach the Mohegan Sun casino when it overturned on Interstate 95 in Madison, east of New Haven. Thirty-six people were hurt in addition to the person who died.
The company's drivers have been cited several times over the past 12 months for safety violations, including failing to obey a traffic control device, speeding and unlawful parking in the roadway.
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