It would be inaccurate to call Tim Berners-Lee's startup a "new internet".
Berners-Lee is now running a startup called Inrupt, which promotes a technology that it is calling Solid. According to the website, Solid is “a technology for organizing data, applications, and identities on the web. [It] enables richer choices for people, organizations, and app developers by building on existing web standards.”
According to WIRED, Solid is a “new web infrastructure called Solid, which rethinks how web apps store and share personal data”. Essentially, Solid will allow individuals to store their own data on a “pod”, located on a Solid server, giving them more control. As such, Solid is not a new internet. Instead, it is a product that lets internet users have more control over what happens with their personal information.