Laura Ingraham went on Sean Hannity’s show on Monday to commiserate with her fellow Fox News host, who was struggling to brush aside the embarrassing revelation that he had taken legal advice from Michael Cohen, President Donald Trump’s beleaguered lawyer. What made the news awkward for Hannity was that he had repeatedly defended Cohen from political critique without acknowledging his relationship with the lawyer. “You’re like my brother, but I’m glad for like a millisecond the heat’s off me and on you,” said Ingraham, whose insults of Stoneman Douglas High School shooting survivor David Hogg sparked a boycott that led nearly 20 companies to pull their ads from her show.

The squabbles surrounding Hannity and Ingraham illustrate the paradoxical position of Fox News in the Trump era: The network is at the pinnacle of power, but that makes it more vulnerable to attacks. What’s true of Fox also applies, for different reasons, to other media outlets on the hard right, like Breitbart, and even outright conspiracy peddlers like Alex Jones’ Infowars. They have achieved unprecedented political impact, but that has brought more intense scrutiny of their journalistic integrity, along with boycotts and lawsuits.

With Trump’s presidency, Fox News has garnered a level of sway in the White House that no other media outlet has ever had. Trump is an avid viewer of the network, and often seems to base his rhetoric and policies on what he sees on the network. Trump has been known to invite Fox hosts to dinner to seek their advice and he’s made major appointments (including national security advisor John Bolton and national economics adviser Larry Kudlow) based on how those figures comported themselves on Fox.

The symbiotic relationship between Fox News and Trump has made the network uniquely consequential, but risks tying the network’s fate to the president. Fox has committed itself to a hardline defense of Trump, with hosts like Hannity and Tucker Carlson pushing with the conspiracy theory that Trump is the victim of a Deep State plot. The long-term risk for Fox is that in becoming so entangled in Trump, they could become discredited if the president goes down.

Fox faces a more immediate problem, too: losing the modicum of respect it has among its mainstream media peers. While figures like Hannity and Ingraham are frankly acknowledged to be ideological warriors, Fox also runs a news desk bound by the same rules of journalistic ethics used by its non-conservative competitors.

By not informing his audience that he had used Cohen’s legal services, Hannity violated a basic tenet of his trade. Mainstream journalists like NBC News’ Chuck Todd and former National Journal writer Ron Fournier regarded the Hannity as a test case of Fox’s professionalism:

But Fox News continues to stand by Hannity. In a statement on Tuesday, the network said, “While FOX News was unaware of Sean Hannity’s informal relationship with Michael Cohen and was surprised by the announcement in court yesterday, we have reviewed the matter and spoken to Sean and he continues to have our full support.” In continuing to offer Hannity “full support” Fox News lent credence to its harshest critics, who contend the network has no journalistic ethics and is nothing more than a propaganda outfit.

Fox isn’t alone in suffering reputational damage, as pro-Trump outlets further to the right are also under scrutiny. Breitbart is widely known as a platform for Trumpism, but the intimacy between the media outlet and Trump is much more extensive than previously expected.

In a report to the British Parliament on Tuesday, Brittany Kaiser, a former employee of Cambridge Analytica, revealed that “a key part of our pitch to Republican clients, was that we had secured exclusive rights to resell Breitbart engagement data. This meant that we had at least some access to what tens of millions of Americans were reading on Breitbart, and could feed this data into our campaign models to help predict resonant issues—and to influence behaviour.”

Given Kaiser’s testimony, it’s hard to see Breitbart as an independent media outlet in any sense. Rather, it was an integral part of the Trump campaign. Breitbart, the Trump campaign and Cambridge Analytical were all spearheaded by Steve Bannon in 2016 as coordinated enterprises. As The Daily Beast put it, they worked “hand in glove.”

Infowars host Alex Jones, long a leading proponent of conspiracy theories, is becoming mired in controversy as well. Two families who lost children killed in the Sandy Hook Elementary School shooting are suing him for claiming the massacre was a hoax.

It’s comforting to believe that Jones is so extreme and outlandish a figure that he can’t be worried about. But in the Trump era, Jones has made genuine entry points into mainstream politics. As a candidate, Trump appeared on Jones’s program, and by Jones’s account, the president still calls him. As CNN’s Andrew Kaczynski has documented, Jones-inspired conspiracy theories have been echoed by high-level GOP officials.

Fox News, Breitbart, and Jones occupy different ideological positions on the American right. Fox is the avatar of the Republican right, Breitbart of the alt-right, and Jones of the conspiratorial right. But they can all take credit for helping to elect Trump. In their triumph, though, they’ve become news stories themselves, a dangerous fate for any media outlet. As Trump has come under fire, so have they. But their critics shouldn’t celebrate quite yet.

While these outlets’ reputations may have been diminished under Trump, and boycotts have hurt one of them, it’s not clear they’re being punished by their fans. Breitbart’s traffic plunge since January may be largely the result of changes to Facebook’s algorithm. Infowars’ traffic has remained steady. Hannity has been neck-and-neck in the ratings with MSNBC’s Rachel Maddow. And after disappearing for a week earlier this month, Ingraham has drawn considerably more viewers since before her controversy over Hogg.

All of which suggests that these outlets will continue to engage in unsavory journalism. The question is whether that will change if Trump goes down. In the meantime, their critics in media are duty-bound not to let up.