US news sites including NPR and PBS left X last year after they were labelled as state-affiliated media under rules introduced by Mr Musk.
Thousands of social media users have quit X in recent days following Mr Trump’s election win to join rival app Bluesky.
Bluesky, which was originally launched by Twitter founder Jack Dorsey as a research project, has added more than 1m users over the past week with the app closing in on 15m users.
Development started in 2019, while Mr Dorsey still ran Twitter, as a “decentralised” version of the app. It was spun-off after Mr Musk’s takeover and officially launched last year.
The app is now primarily owned by Jay Graber, its chief executive, and operates as an independent company. In October it raised $15m (£12m) from investors Blockchain Capital.
Bluesky is currently ranking top in the US iPhone App Store, while it is fifth in the UK store.
Threads, a rival service with a similar design to X that is owned by Mark Zuckerberg’s Meta, has swelled to 275m users since it launched in July last year.
The Guardian does not have a main Bluesky account, although its primary Threads account has 1.3m followers. Its main X account has more than 10m followers.
Last week, it emerged that The Guardian had offered staff counselling in the wake of Mr Trump’s US election win. Katharine Viner, the newspaper’s editor, said the result would be “upsetting” for many staff.
A spokesman said this was part of its standard “employee assistance programme”, which is available at all times.
Mr Musk, the world’s richest man, spent election night with Mr Trump in Mar-a-Lago and campaigned for the Republican for weeks. Mr Musk claimed last week that X had enjoyed “record usage” during the US election.
On Tuesday, Mr Trump confirmed the Tesla boss would take the helm of a new Department of Government Efficiency to enact sweeping cost cuts in the US.
Mr Bailey will say the changed relationship with the EU has "weighed" on the economy."The impact on trade seems to be more in goods than services... But it unde
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Rachel Reeves has told City bankers attending her Mansion House speech that regulations put in place to protect the economy after the global financial crisis ha