"I know they can't make this country look like Sweden," said Adnan Besirovic, "but we just want them to do their job. They drive BMWs with tinted windows and they have big salaries, the biggest in the region, but they don't do anything."

The people he refers to are Bosnia's politicians, who are coming under unprecedented pressure to deal with a mood of rage that has tilted the country into its worst scenes of violence since the civil war ended in 1995.

"We have had 20 years of bad leadership," added Besirovic, a 29-year-old film producer.

The protests that began spontaneously in the northern town of Tuzla a week ago have quickly spread to the capital, Sarajevo, and other large urban areas, drawing in workers, students and war veterans with a wide range of grievances.

A protester in Tuzla, Bosnia. Photograph: Amel Emric/AP

The protesters have become organised, rallying behind a series of common grievances, focused on dissatisfaction with the entire political class and Bosnia's cumbersome, dysfunctional and disproportionately expensive government structure. There are even signs that the protests could jump Bosnia's postwar boundaries and spread into the Serb-run half of the country.

The multi-layered system of government for the Federation of Bosnia and Herzegovina was an attempt to balance ethnic rivalries after the 1995 Dayton peace agreement, the deal that ended more than three years of war and "ethnic cleansing" when more than 100,000 people were killed. The Bosnian Serb leaders Radovan Karadžić and Ratko Mladić are at present on trial for genocide, perpetrated during this period, at the Hague war crimes tribunal.

The post-Dayton system has become impossibly unwieldy. Each of the 10 federal cantons has a prime minister and cabinet. Governments compete in the federation. Then there is the Republika Srpska (Serb republic) and a weak Bosnian national government, which is supposed to coordinate all the layers below it.

"I don't want ministers. I don't want cantons. I don't want 15 governments. Ministers are morons," read one of the banners at the Sarajevo demonstrations.

The protesters have been spurred on by their success so far in forcing the resignation of governments in four of the federation's cantons.

In Sarajevo, a protest meeting was called for Wednesday to agree a new platform of demands. Nermin Nikšić, the federation prime minister, has shrugged off calls for his government to resign, but under the increasing pressure of protests has offered to call snap elections.

A few thousand protesters gathered on Tuesday at a central crossroads in Sarajevo near the burnt-out cantonal and municipal government buildings that were attacked in riots over the weekend. The main theme of their banners was opposition to government overspending and corruption amid chronic high unemployment and a stagnant economy.

Arnela Dizdarevic, a 21-year-old sports student, held up a sign that said: "There is far too much government but no salaries, no jobs and no health insurance. We are going to stay here and keep coming back until something changes. My father was in the frontline to defend this country in 1992. If I can be there to support my country, my people, I will be."

The crowd of demonstrators reflected a widely shared belief that a self-serving political elite from all parties had siphoned off billions of dollars of aid money and investment capital over 20 years. A feature of the postwar period has been fraudulent privatisations of public enterprises, which have subsequently been stripped of assets and allowed to collapse. The bankruptcy of a string of such companies in Tuzla triggered the wave of protests.

Marko Vesovic, a writer and poet, spent four years as a member of parliament but now supports the protests against a corrupt political class.

He said of his former parliamentary colleagues: "I watched what they were doing. It went on right in front of my eyes. My wage was 3,000 marks [about £1,270] a month. I felt like a duke. But I realised it was just pocket money for the other MPs. What they rob from the state is much more, and the stronger the position in your party the more you can steal."

Although the majority of the protests have so far been in mostly Bosniak (Muslim) areas, Vesovic rejected the suggestion of there being an ethnic dimension to the demonstrations. "In Bosnia, finally, the nationalist narrative of the political parties has been swallowed by social narrative on the streets," he said. A banner at the demonstrations mocked Bosnia's ethnocentric politics, declaring: "I am hungry in three languages."

In an attempt to prevent the unrest spreading to his territory, the Republika Srpska president, Milorad Dodik, has sought to portray it as a Bosniak-Croat plot aimed at Serbs – a claim echoed by the mostly pliant Serb media. The Republika Srpska Press newspaper claimed in a headline: "Demonstrators given weapons to go to attack Republika Srpska", although there was no evidence of any such conspiracy.

Dodik's strategy has failed to stem the gradual growth of protests in Serb towns, including in Prijedor, Banja Luka and Brčko. Prijedor's demonstrators produced a list of 10 demands, including "the resignations of all the directors of public institutions" in the town, scene of some of the worst atrocities of the 1992-95 war.

The Bosnian Serb war veterans' association has ridiculed Dodik's insistence that all means would be used to prevent an uprising in the Republika Srpska. The association declared: "These statements will only stoke flames that have been lit by those in power, who are attempting by any means necessary to preserve a state that is based on crime, corruption, nepotism, and on a horrendous education system whose consequences are already being felt."