London's FTSE 100 index of leading shares closed up nearly 1.77%, while the main markets in France and Germany rose 1.97% and 1.94%.
Confidence was boosted when the head of Europe's central bank promised action to steady the eurozone if necessary.
It offset fears about the falling oil price and worries about global growth.
In the US, shares also recovered from losses the previous day, with traders saying that some investors believed the market was over-sold.
About half-way through Wall Street's trading session, the Dow Jones was up 1.54% and the S&P 500 was 1.46% ahead.
Much earlier, Japan's main share index closed down by more than 2%.
On Wednesday, global stock markets suffered hefty losses and London's FTSE 100 ended the day down 3.5%.
By doing so it entered a "bear market", having fallen 20% from its record high in April last year.
Comments by European Central Bank president Mario Draghi helped to steady investors' nerves.
He hinted that the ECB could do more to stimulate the eurozone economy, saying there were "no limits" to action if necessary.
The oil price also recovered, although it remains at around 12-year lows.
Brent crude rose 5.9% to $29.52 a barrel. In the US, West Texas Intermediate Crude rose 5.3% briefly breaking back above $30 a barrel before settling at $29.79.
Oil prices have been falling since mid 2014, but oil-producing countries have maintained output despite the decline, contributing to the excess supplies on the market.
Earlier in the week, the International Energy Agency warned that oil markets could "drown in oversupply" in 2016.
Patrick Thomson from JP Morgan Asset Management told the BBC that investors should not panic.
"If you look at the US economy particularly, that is actually in pretty good shape," he said.
"You look at all of the data coming out recently, clearly growth is a little muted and corporate earnings are somewhat lower than expected due to energy prices and the strong dollar, but underlying fundamentals, particularly the US consumer, is in very good shape."
That message was echoed by analysts Capital Economics, which said: "Despite the prevailing gloom about the world economy, we think global growth will pick up from around 2.5% last year to 3% in both 2016 and 2017, using our own estimates for China."
Europe's stock markets bounced back on Thursday, a day after billions were wiped off the value of shares amid global market turmoil.