The history of Looney Tunes cartoons on TV is...complicated. WB sold the pre-1948 cartoons off to separate companies (Associated Artists Productions for the color cartoons, Sunset Productions for black and white) while keeping the rest for themselves. These post-1948 cartoons ran heavily on Saturday mornings for decades, mainly the more popular ones with the rest running in syndicated packages. The pre-1948 stuff ran in separate syndication packages well before the newer stuff.
By the 80s, the dynamic had changed a bit. The B&W cartoons had been acquired by a company called Seven Arts Productions, which merged with WB in 1967, putting those cartoons back in the WB fold. AAP, meanwhile, was bought out by United Artists, which was itself bought out by MGM after Heaven's Gate bombed. That library was taken by Ted Turner during his short-lived ownership of the combined company, and when he sold the whole lot back, he kept much of the library, including the color Looney Tunes. As for WB's own packages, in the 90s they were split between ABC on Saturday mornings (I remember this particularly well), Nickelodeon, and a weekday package of cartoons that was initially syndicated, then moved to Fox before becoming part of Kids WB's initial lineup. The Turner package, meanwhile, became a linchpin for the launch of Cartoon Network in 1992.
Then, Time Warner bought Turner, bringing all the Looney Tunes cartoons together under one roof. The Kids' WB and CN packages were immediately merged together, and from there, WB started pulling the shorts from all non-CN sources. Nickelodeon dropped its package in 1999, Kids' WB phased out the shorts as the Pokemon craze was in full swing, and ABC ended its LT show in the fall of 1999, and the shorts would remain exclusive to the Turner networks from that point on until earlier this year, when MeTV began running the shorts.
TL;DR - WB sold off the old shorts, then got them back through a series of corporate buyouts. They used to be everywhere, but they were relegated to CN and Boomerang in the new millennium.