The most oft-cited example of American influence on extremism (I'll assume you're speaking of the Islamic variety; "extremism" is also a very broad and problematic term), at least by way of the Cold War, revolves around the Soviet invasion of Afghanistan in 1979. The Marxist People's Democratic Party of Afghanistan (PDPA) had just overthrown and executed the secular third-worldist government of Mohamed Daoud Khan, which had been seeking to distance Afghanistan from both Islamist politics and its reliance upon the Soviets. Now facing a potential civil war with Islamist opposition (mujahideen) that had been excluded from the takeover, the PDPA requested that the USSR step in to ensure that Communists would retain their exclusive control of Afghanistan.

Before and during this critical time the United States thus looked to counter Communist influence in the region by arming the mujahideen and stirring up resentment among Afghanistan's very large religious constituency, politicizing Islam for the purpose of countering Communist ideology. They sought to spread the idea of "Godless Communism" in an Islamic context, emphasizing that accepting Communist rule meant that Afghans would have to give up their deep religious tradition and even be forced to become atheists like the Russians. So to this end and beyond the military aid, the Americans funded extremist madrassahs (Islamic schools) and teachers throughout the countryside and provided them with USAID-made textbooks emphasizing the duty of Muslims to violently defend the Islamic community against non-Muslim invaders. Owing to the limited means of the Afghan education system, many of these textbooks are still in use today. They're furthermore embraced and reprinted by jihadis who have turned that same line of thinking against Western secularism and interventionism.

The US was also more than willing to tolerate and/or implicitly support Osama bin Laden's formation of the Arab Afghan brigades at this time. He recruited hundreds of fighters and raised millions of dollars from around the world, including from the USA itself, to support the jihad in Afghanistan. There's little evidence of direct funding from the US government to the Arab Afghans, but the Arab Afghans ultimately received many indirect benefits from US policy towards their mujahideen allies. Another major factor is that U.S. policy towards Saudi Arabia has always tolerated Wahabbi extremism and Saudi Wahabbi support for movements such as the Arab Afghans.

The Arab Afghans were comparatively small in number and did little to help the mujahideen aside from moral and ideological support. But they would grow in power and influence after the Soviet withdrawal. After a brutal civil war between the Islamist factions who drove out the Soviets, the Taliban eventually united to defeat the warring factions and rampant organized crime with an iron fist. The Arab Afghans meanwhile went on to found al-Qaeda. You probably know the rest.

Although it frequently avoids or sugarcoats the USA's involvement in empowering bloodthirsty Islamist warlords, I'd recommend reading or even just watching Charlie Wilson's War for an entertaining USA-centric view of this history.