Just because a government exists does not mean we don't live under capitalism. 90% of the GDP of the United States comes from the private sector (source). Non-profits only make up about 5% of the GDP (source). Therefore, about 85% of the US economy is private, for-profit business. In other words, the US is overwhelmingly a capitalist nation.
Regardless, the government has everything to do with capitalism. Even though the government doesn't much trade and industry under capitalism, it still is crucial to the capitalist system because it enforces legal rights to private property. Without the government, private property rights would not be enforced, and therefore private property would not really exist, at least not in the way it does now.
Furthermore, in democratic government that have privately-funded elections are greatly influenced by private business. Through private campaign contributions and promises of sinecures once they retire from public service, democratic politicians are incentivized to act in ways which please private business if those politicians want to be reelected. Even ignoring politicians' own incentives, private business is incentivized to support, through campaign contributions, politicians whose policies align with their interests. This effects how a nation behaves, including in the international arena. Politicians are incentivized to go to war if business interests, especially those that benefit from access to natural resources and open foreign markets, can profit from war, and those business are incentivized to support warmongering politicians. As such, private business, and therefore capitalism, is partially responsible for the mess that currently exists in the Middle East, including Syria.