You actually mention one of the reasons why in your post- Rush Limbaugh. Or, more specifically, the industry that he represents, right-wing talk radio.

Up until 1987, the FCC had a policy in place called the "Fairness Doctrine." Essentially, if you broadcast an opinion on a controversial issue, you were supposed to allow the opposing position to also air on your station. The idea was to provide a balanced discourse on topics that were in the public interest.

This doctrine had been upheld as constitutional in a variety of Supreme Court cases, although the decisions were usually careful to note that A) the doctrine should not pose undue hindrances to the right to free speech and B) though the doctrine was constitutional, the court didn't find it necessary, and would make no ruling on whether removing it would be unconstitutional.

This didn't matter, though, when the FCC (under the Reagan administration) did away with the doctrine in 1987. Broadcasters saw an opening, and the ensuing decade saw the rise of talk radio around the country, particularly with a conservative slant. Limbaugh got his national show in the summer of 1988, and his show was soon recognized as one of the most-listened-to programs on the air nationally.

So, at least part of your answer can come from simple timing. Up until 1993, the US was under a Republican president. Clinton was the first Democrat to come into office as conservative talk radio became a juggernaut, and so became the primary target for his time in office.

Bush's (Sr's) later term was marked by cracks showing up in the Reagan coalition. This coalition had successfully crushed Carter, Mondale, and Dukakis, but by 1991 (effectively 11 years of having the same coalition and party in power). Bush successfully ramped up the operation against Iraq, fought the Gulf war, and pulled out of Iraq, but despite all this, he had a lot of trouble on the home front. Namely pushing a series of much criticized tax increases in 1990 to balance the budget, which kind of went against his "no new taxes pledge". This really didn't go over well with a large part of his base, and, while he won the nomination of his party, he had to defend himself in the primaries against Pat Buchanan (which for a sitting president to now have to do that looks pretty bad). The notion of Bush being a "betrayer" to the conservative cause and not worth releecting sent chaos within the Republican ranks with, and this chaos also fueled exterior parties such as the Reform Party under Ross Perot to unexpectedly emerge and poll so well that he was given a seat at the presidential debates.

With all this chaos occurring both within and outside of the Republican Party, the Democratic Nomination also occured. And while it was a heated battle at first, Clinton's overwhelming popularity with Southern democrats helped him secure some big wins, and the nomination snowballed into a Clinton landslide. Once that was complete Clinton was the unquestionable leader and face of the Democratic Party. With the effects of an economic recession, infighting in his party, and Ross Perot upset Bush was unable to cobble enough voters to his side unlike Clinton who cemented the Democratic base together in order to win him a plurality of the popular vote, but a massive electoral victory emerged.

With this Clinton was the first Democrat in power since Carter, and Bush withdrew as head of the party and the Republican party overall withdrew to lick their wounds. In the midst of a leadership-less party, the one thing that united the party was opposition to Clinton, and I should say Clintons given how high of a profile Hillary sought and maintained once Bill went into office. With the loss of the White House the conservative media (as DGBD explains) went into overdrive in basically opposition to all things Clinton as the unifying strategy of the party after 1992. This strategy (hammering the Clinton's on radio and tv) proved effective as the Democrats stumbled to cobble together a healthcare bill and ultimately failed during their efforts 1992-3 (Obamacare would the next time something like that would be attempted).

Gingrich emerged as something of a party leader with his "Contract with America" plan, and his efforts to unify the party under his principles and against Clinton paid off by delivering the House to the Republican Party (a first since 1954). Seemingly the campaign to just utterly attack the Clintons on all fronts paid off, and the party continued to do that strategy, but not enough to put together a presidential candidate to unseat Clinton in 1996. And since the much remembered events of the latter Clinton years very much fall under the 20 year rule I'll leave this at that.

/u/The_Alaskan wrote a characteristically excellent response in this thread about Hillary.