Some say Brazilian President Jair Bolsonaro is having a bad stretch, after ditching his two most popular ministers and going far out on an anti-confinement limb in the middle of the coronavirus crisis.

But the far-right president's hard-core supporters are loving his battle against the "establishment."

By many measures, the last few weeks have been tough for Bolsonaro, the man some call the "Tropical Trump" for his outsider style, politically incorrect vitriol and the love of social media they both share.

First came his controversial decision to fire health minister Luiz Henrique Mandetta in mid-April, after weeks of publicly fighting with him over the coronavirus.

Mandetta, who led the charge for aggressive social distancing measures in Brazil, was highly popular.

Jair Bolsonaro speaks at a press conference on April 24 in Brasilia
Jair Bolsonaro speaks at a press conference on April 24 in Brasilia AFP / EVARISTO SA

And Bolsonaro, who compares the virus to a "little flu," is increasingly isolated at home and abroad for his insistence that containment measures are an overreaction, even as the death toll has surged past 5,000 in Brazil.

Then came Bolsonaro's decision last week to fire the head of the federal police, which caused the also highly popular justice minister, Sergio Moro, to resign in protest.

In a scathing final press conference, Moro accused Bolsonaro of "political interference," leading a Supreme Court judge to order a potentially explosive probe into whether the president committed obstruction of justice to protect himself and his family from a series of ongoing investigations.

Political analysts have started talking seriously about the possibility Bolsonaro could be impeached before he even gets a chance to seek re-election in 2022.

Despite all that, when the latest opinion poll from the respected Datafolha institute came out this week, conducted three days after Moro's exit, there was no dent in Bolsonaro's support.

Sergio Moro is seen on April 24 in Brasilia
Sergio Moro is seen on April 24 in Brasilia AFP / EVARISTO SA

In fact, it rose slightly, to 33 percent, up from 30 percent in December and 32 percent a year ago when he was a few months into his term.

"Bolsonaro's core voters are fed up" with politics as usual in Brasilia, said political analyst Murillo de Aragao.

"He dares to say what many Brazilians are thinking."

As the coronavirus crisis has deepened, Bolsonaro has faced nightly protests in many middle-class and wealthy neighborhoods, where residents bang pots and pans out their windows and shout, "Get out, Bolsonaro!"

Luiz Henrique Mandetta attends a press conference on the coronavirus situation on March 18 in Brasilia
Luiz Henrique Mandetta attends a press conference on the coronavirus situation on March 18 in Brasilia AFP / Sergio LIMA

But protests are less common in poor neighborhoods, where people often need to leave the house to earn enough money to feed their families.

And it appears there is a hard core of supporters for whom the president can do no wrong. Look no further than his middle name -- "Messias," or messiah -- for evidence.

Breaking with his justice minister was a test of how solid that core is.

To many voters to the right or center in Brazil, Moro is an anti-corruption superhero.

Before he joined Bolsonaro's government, he was a federal judge who shot to fame for presiding over "Operation Car Wash," a massive anti-corruption investigation that brought down a long list of powerful politicians and businessmen, including ex-president Luiz Inacio Lula da Silva.

De Aragao said Moro's exit likely "shook" Bolsonaro's popularity, and may yet have a bigger impact in subsequent polls.

But "Brazil is a conservative, religious country," said Ivar Hartmann, a law professor at the Getulio Vargas Foundation.

Bolsonaro is a conservative Catholic with strong support from Brazil's burgeoning Evangelical community.

Sixteen months into his term, Bolsonaro is still hammering home his anti-establishment message.

And it seems to be working for him, despite -- or perhaps because of -- mounting criticism from powerful figures in Congress and state governments, one-time supporters in the business world, and in the opinion pages of the country's leading newspapers.

"If a powerful group of businessmen had withdrawn their support for the government 20 or 30 years ago, plus a major media organization, that would have had an impact," said Hartmann.

"But today, there's a growing disconnect between the people and the press, especially for that one-third of the population that relies on pseudo-news sites."

Thanks to social media, the president reaches that public directly, "with no filters," he added.

But with tension mounting between the president, the courts and the Congress, "we are facing an institutional war," said de Aragao.