It's...complicated. Since the 1970s/1980s Saudi Arabia has certainly had a policy of funding mosques and mosque construction, not just in Europe but around the world. It's not much of an exaggeration to say that if you live near a mosque that was built in the past 50 years that it probably had some degree of Saudi funding in its construction, and even older mosques might well still receive Saudi funding for their operation. In the 1980s, for instance, some 800 mosques in the UK were Saudi-funded.

Sometimes this funding is quiet. To give a recent (within 20 year) example of longstanding policy, the Saudi connection to the Islamic Society of Boston had to be discovered by reporters. At other times, and particularly in Islamic countries, the donations/funding are an explicit diplomatic tool to demonstrate goodwill. That aspect of Saudi policy predates the usually marked turning point for Saudi policy of 1979, as in, for example, the construction of the Faisal Mosque in Pakistan (begun in 1976), but since 1979 that amount of money being spent grew enormously as part of a deliberate policy shift.

Sometimes this money is donated directly by the state, sometimes by members of the royal family or government, and sometimes through private charity networks. Accusations of various things being "Saudi funded" rarely distinguish between these different forms of payment. The extent to which the distinction matters is blurred in a society where wealth, the state, and the royal family are so closely interlinked.

In the case of "radical" Islam specifically, for instance, the 9/11 commission report describes al-Qaeda's funding as follows:

Al Qaeda appears to have relied on a core group of financial facilitators who raised money from a variety of donors and other fund-raisers, primarily in the Gulf countries and particularly in Saudi Arabia. Some individual donors surely knew, and others did not, the ultimate destination of their donations. Al Qaeda and its friends took advantage of Islam's strong calls for charitable giving, zakat. These financial facilitators also appeared to rely heavily on certain imams at mosques who were willing to divert zakat donations to al Qaeda's cause.

Al Qaeda also collected money from employees of corrupt charities. It took two approaches to using charities for fund-raising. ONe was to rely on al Qaeda sympathizers in specific foreign branch offices of large, international charities—particularly those with lax external oversight and ineffective internal controls, such as the Saudi-based al Haramain Islamic Foundation. Smaller charities in various parts of the globe were funded by these large Gulf charities and had employees who would siphon the money to al Qaeda.

In addition, entire charities, such as the al Wafa organization, may have wittingly participated in funneling money to al Qaeda. In those cases, al Qaeda operatives controlled the entire organization, including access to bank accounts. Charities were a source of money and also provided significant cover, which enabled operatives to travel undetected under the guise of working for a humanitarian organization.

It does not appear that any government other than the Taliban financially supported al Qaeda before 9/11, although some governments may have contained al Qaeda sympathizers who turned a blind eye to al Qaeda's fundraising activities. Saudi Arabia has long been considered the primary source of al Qaeda funding, but we have found no evidence that the Saudi government as an institution or senior Saudi officials individually funded the organization. (This conclusion does not exclude the likelihood that charities with significant Saudi government sponsorship diverted funds to al Qaeda.)

Still, al Qaeda found fertile fund-raising ground in Saudi Arabia...

That of course is for a Salafist-Jihadist organization explicitly dedicated to carrying out or facilitating international terrorism.

One of the issues with your question is how we define "radical" Islam. Whatever a "radical" is, al Qaeda certainly qualifies. But who else? I, for one, find the term to be vague and not very useful. I would probably apply to groups that explicitly carry out violence against civilians, or to preachers who espouse the same, but I would be hesitant to apply to all violent Islamist groups. Were the Afghan mujahideen fighting the Soviets "radicals" for instance? Would a preacher advocating armed resistance against Saddam Hussein's invasion of Kuwait have been "radical"?

Generally I prefer typological terms like Islamist, Salafist, Jihadist, etc. These terms are themselves not without controversy. For example, in general I wouldn't describe the government of Saudi Arabia as an "Islamist" one, insofar as Muslim Brotherhood-style Islamism is a revolutionary ideology, whereas the state ideology of Saudi Arabia is status quo, even if what the Muslim Brotherhood theoretically wants is a state that more closely resembles Saudi Arabia than, say, Nasser's Egypt.

I don't think it would be worth the time here to discuss the ways in which what the Saudis have exported internationally is or is not Islamist, Salafist, Jihadist, or Radical. I think it would probably be easier to simply conclude that in general what they have exported is a kind of Islam more closely resembles Islam as it is practiced in Saudi Arabia. The state version of this is "Wahhabism", after the effective co-founder of the First Saudi State, Muhammad ibn abd al-Wahhab. I've written before some of the particular tenets of Wahhabism that may be of interest HERE. In short, however, it is a kind of Salafism, that is, it bases its claims to religious authority on direct appeals the writings, views, and practices of Muhammad, his companions, and the two generations that succeeded the companions, who are collectively "the Salaf."

This is in contrast with traditional Islam as practiced in, say, the Ottoman Empire, which had supplemental bodies of legal codes and jurisprudence, traditions, and non-textual practices like Sufism (Islamic mysticism) as well as an organized scholarly tradition controlled through institutions like Egypt's al-Azhar (traditionally a madrasa, now a university.)

The Saudi export of their version of Islam has come in several forms. The funding of mosques is one aspect. But the Saudis also produce educational materials and themselves educate imams who are then sent overseas to preach in these mosques.

Perhaps more important in the spread of more violent or radical messages in the 1980s and 1990s however were cassette tapes and CDs. Cassette tapes allowed sermons from violent or politically radical preachers to circulate even in countries where what could be openly preached was restricted by the state, as in Mubarak's Egypt. Osama bin Laden when he was killed in Pakistan was found to have a collection of 1,500 cassette tapes, most of which were sermons and speeches.

Saudi Arabia itself has over time exercised increased control over preaching in the Kingdom. This was especially important in relation to the Sahwah movement, which was partly the result of a combination of traditional Saudi religious views with a more Brotherhood-inspired political orientation. This group's [edit: they're perhaps better described as a "trend" or movement than a group] opposition to the US-presence in Saudi Arabia and stance on the Gulf War was a key influence, for example, on Osama bin Laden, who took similar views but ended up at a more violent conclusion.

If I can make this rambling attempt at an answer even more rambling I think the other key question is to what extent are these Saudi elements somehow uniquely responsible for violent Islamist extremism in Europe and around the world. That's a difficult question to answer but I think you have to look at other elements, both ideological ones, like the influence of various Muslim Brotherhood thinkers like Sayyid Qutb and of the Brotherhood's various offshoots like Hamas, as well as what are sometimes called "root cause" political issues, whether that be European discrimination and ghettoization of Muslim immigrants, or conflicts driving jihad like the Soviet invasion of Afghanistan.

This is to a point getting into 20 year issues but there's also the extent to which Saudi Arabia ceased or took control of this bad behavior. Certainly the wild west of Islamic charities with money going from private hands uncontrolled to various international terrorist groups has been cracked down upon since 9/11, although the effectiveness of this is debated. Saudi Arabia has also developed a serious counter-terrorism apparatus and cooperates closely with Western countries. But in a sense as we all know the cat is out of the bag. Whatever the Saudis were doing in the 1980s, they could halt everything now and transnational jihad would still be self-sustaining given the ability for individuals to self-radicalize and carry out attacks on their own initiative for the cost of a car rental or a machete.

Edit: couple minor changes, I still have some typos in here I might try to fix :) Edit 2: Also happy to take any followup questions anyone might have as best as I can.