The savage suicide bombings in the heart of Baghdad yesterday show how far the violence in Iraq is from being over. It is as if those who order these bombings know that they only have to repeat these atrocities every couple of months to destabilise the country.

The government of Prime Minister Nouri al-Maliki makes itself even more vulnerable by boasting that it is improving security. Iraq is a safer place than it was three years ago, but it is still one of the more dangerous places in the world.

There is no need to imagine that the slaughter in Haifa Street yesterday was because American troops withdrew from the cities of Iraq three months ago. With or without US troops, the bombers have been able to get through in Baghdad ever since they destroyed the UN headquarters in 2003.

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Suicide car bombings, even when the driver is not planning to detonate his deadly cargo personally, are extremely difficult to stop. Remember the success the Provisional IRA had in the 1990s in targeting much smaller areas in the city of London and Canary Wharf.

After a bomb eviscerated the Iraqi Foreign Ministry on 19 August the Foreign Minister Hoshyar Zebari said its passage must have been helped by collaborators at army and police checkpoints. This may be true. But it is impossible for Iraqi security to search every vehicle, especially as bombers will have made sure that their papers are in order. It will also have occurred to Iraqi soldiers and policemen that any awards for stopping a suicide bomber are likely to be posthumous. Enthusiasm for investigating suspicious vehicles is limited.

The bombings do not by themselves prove that Iraq remains unstable. Unfortunately, there are other pointers such as the failure of 1.6 million internally displaced people to return to their homes. A study by the International Organisation on Migration explains why these internal Iraqi refugees are not going home. It says that security may have got better but refugees are still trying to survive "without work, their own home, schooling for children, access to water, electricity and health care".

Who is behind the bombings? Almost certainly it is some cell of al-Qa'ida, possibly acting with the guidance or help of the Baath party or the security service of the old regime. Al-Qa'ida is not as strong as it was in 2007, but then it does not have to be to create mayhem.

The main problem in Iraq is that there is no fundamental agreement between the three main communities: the Shia, the Sunni Arabs and the Kurds. Each group is still looking for the weak points of the others. The Shia are three-fifths of the population, benefited from the overthrow of the predominantly-Sunni regime of Saddam Hussein and were largely victorious in the sectarian battle for Baghdad in 2005-7. This does not mean that the Sunni, who make up a fifth of the population, do not retain the strength to destabilise the government unless they get the share of power that they want.

Iraqis themselves tend to see the never-ending violence as a sign that their neighbours intend to prevent the re-emergence of a strong Iraq. Iran would like another Shia state in the Gulf, but it does not want a powerful government to resurrect itself in Baghdad. Saudi Arabia has long been aghast at seeing Iraq becoming the first Shia government in the Arab world since Saladin overthrew the Fatimids. Kuwait is still taking part of Iraq's desperately needed oil revenues in compensation for its losses in the first Gulf War.

A further problem is Iraq's undermining the political and economic reconstruction. The country has had 30 years of war, rebellion and economic sanctions. Iraq truly is a broken society. The state is dysfunctional. There is some good news: the price of oil has risen to $80 a barrel. But even relatively peaceful cities like Basra are full of people who are not being paid. The government is failing to heal the deep wounds of the past. Yesterday's bombings – the deadliest in two years – shows how far Iraq is from solving its problems.