I don't know what you mean by 'the real London bridge is not the current one'. What??? Sorry, but that makes no sense in any language!
If you mean 'Is it true that the fancy bridge over the Thames with the elaborate towers isn't in fact called London Bridge?' then yes, it's true. The iconic image of London, which you see on all the postcards and souvenirs, is Tower Bridge. It's famous, but not particularly old - it was built at the end of the 19th century.
London Bridge is to the west of Tower Bridge. It's called London Bridge because was the first one to to be built. There's been London Bridge since Roman times, 2000 years ago, but it has been replaced on a number of occasions. The current one is a plain steel and concrete construction, opened in 1973.
The previous London Bridge, a Victorian stone arched bridge, was taken apart stone by stone and rebuilt in Arizona in the USA. There's a myth that the Americans bought it by mistake because they thought that they were getting the fancier Tower Bridge! Probably not true, though.