The average speed of home broadband is now 12Mbps (megabits per second), according to research from regulator Ofcom.
That has risen by a third since its last report in May 2012 and trebled in the past four years.
Increasingly consumers are upgrading their broadband packages to super-fast services of 30Mbps or above.
The increasing appetite for online video and more devices per household were driving the changes, Ofcom said.
By November 2012, more than three-quarters of the UK's home broadband users were on packages with advertised speeds above 10Mbps.
The proportion of broadband connections classed as superfast was up to 13%, from 5% the previous year.
Superfast connections are getting faster, with average speeds rising from 35.8Mbps in May 2012 to 44.6Mbps in November 2012.
They are also getting cheaper - customers can upgrade for as little as £5, according to Ofcom.
Ed Richards, Ofcom's chief executive, said "Our research shows that UK consumers are adopting faster broadband packages to cater for their increasing use of bandwidth-heavy services such as video streaming.
"The increase in the average number of connected devices in UK homes is also driving the need for speed," he said.
The report also looked at upload speeds, important to consumers wishing to share large files or use real-time video communications. The average upload speed was 1.4Mbps, up from 0.3Mbps average in May.
Price comparison website broadbandchoices.co.uk said that the prices of super-fast services needed to fall.
"An increase of 3.1Mbps (34%) in UK broadband speeds is significant, but this is entirely down to the increased availability of fibre optic services and these do not come cheap - you could end up paying five times as much for fibre optic as you would for standard ADSL," said Dominic Baliszewski, the website's telecoms expert.
He also pointed out that the increasing speeds were unlikely to affect those struggling on much slower speeds in the countryside: "Unfortunately this report will be cold comfort to rural dwellers stuck in 'digital ghettos' who are struggling to get more than a few megabits per second.
"British households are still having to play a postcode lottery when it comes to broadband speed; the government's vision for UK broadband to be fastest in Europe by 2015 is a long way from being realised," he said.
This week the Community Broadband Network launched a report looking at fresh ways to fund broadband to the last 10% of the UK.
It wants the government to provide loans directly to communities wanting to build their own networks.
Meanwhile a report from telecoms analyst firm PointTopic said that there was a natural limit to the speeds consumers would need.
"Today that 'point' is in the 60-70Mbps range," said Oliver Johnson, chief executive at Point Topic.
"Enough to stream a high definition video or three with perhaps some light browsing on the side and if the kids are old enough an online game and a music stream," he added.