how does the government promote economic growth in capitalist countries and those transitioning to capitalism?

In capitalist countries (and those transitioning to capitalism), overt or direct government control of the economy is avoided. Instead, the government seeks to provide the infrastructure and security/stability needed in order for business interests to flourish and the economy to grow.

Direct government control of the economy was the pattern of "mercantilism," which was dominant prior to the rise of capitalism. According to Adam Smith, one of the founding theorists of capitalism, mercantilism was bad because the government tried to regulate and control commerce too much. Smith, along with French economic theorists known "physiocrats" (wanting to let "nature rule"), advocated a "laissez-faire" approach to commerce and industry, letting businesses run themselves in competition in a free market.

However, for that free market to flourish, governments will do their part by providing infrastructure - - roads, ports, etc. Governments also will ensure the stability and security needed for businesses to operate safely and consistently. So governments will enact laws affirming rights and fairness in business contracts and enforce those laws, and also provide national security within their lands so that economic growth can flourish due to peaceful circumstances. (Warfare causes great disruption to economies.)