He did actually walk across the frozen Bering Strait. First attempt was 2005, but he failed. Second attempt in 2006 succeeded
The pair had to take a roundabout 150-mile route to cross the 58-mile wide strait from Alaska to Siberia, crossing sheets of ice which were drifting with the current.
Andrew Cooper, of Goliath Earth Trek UK, said: "The last 14 days have been the equivalent of running a marathon every day, so Karl will take a few days to get some energy back before he sets off again."
And from Wikipedia
Between 17 and 31 March 2006,^^ Bushby and French adventurer Dimitri Kieffer successfully crossed the Bering Strait on foot, having to take a roundabout 14-day route across a frozen 150-mile (240 km) section to cross the 58-mile (93 km) wide strait from Alaska to Siberia.^^ They were detained by Russian border troop officers while they were crossing the Russian border near the Chukotkan village of Uelen, for not entering Russia at a correct port of entry. They were threatened with being banned from Russia, which would stop the journey. It was announced on 5 May 2006 that the Russian appeal court had upheld Bushby's application and his walk would continue