Turkish users of Twitter, including the country's president, have flouted a block on the social media platform by using text messaging services or disguising the location of their computers to continue posting messages on the site.

In what many Twitter users in Turkey called a "digital coup", Telecom regulators enforced four court orders to restrict access to Twitter on Thursday night, just hours after the prime minister, Recep Tayyip Erdogan, vowed to "eradicate" the microblogging platform in an election speech.

The disruption followed previous government threats to clamp down on the social media and caused widespread outrage inside and outside Turkey. In a first reaction to the ban, Neelie Kroes, vice-president of the European commission, tweeted: "The Twitter ban in #Turkey is groundless, pointless, cowardly. Turkish people and intl community will see this as censorship. It is."

Štefan Füle, EU commissioner for Enlargement and European Neighbourhood Policy, said in a statement: "The ban on the social platform Twitter.com in Turkey raises grave concerns and casts doubt on Turkey's stated commitment to European values and standards."

The hashtag #TwitterisblockedinTurkey quickly rose to the top trending term globally. According to social media agency We Are Social the number of tweets sent from Turkey went up 138% following the ban.

Shortly after the Twitter ban came into effect about midnight, the microblogging company tweeted instructions to users in Turkey on how to circumvent it using text messaging services in Turkish and English. Turkish tweeters were quick to share other methods of tiptoeing around the ban, using "virtual private networks" (VPNs) – which allow internet users to connect to the web undetected – or changing the domain name settings on computers and mobile devices to conceal their geographic whereabouts.

Some large Turkish news websites also published step-by-step instructions on how to change domain name system (DNS) settings.

On Friday, Turkey woke up to lively birdsong: according to the alternative online news site Zete.com, almost 2.5m tweets – 17,000 tweets a minute – have been posted from Turkey since the Twitter ban went into effect, setting records for Twitter use in the country. "Boss, my bird is still tweeting… @RT_Erdogan," posted @Fakir_Bey. "And yours?"

But it was not just critics of the government who took to Twitter after the site was closed. The mayor of Ankara, Melih Gökcek, famous for his extensive and rather bullish use of the micro-blogging site, was the first politician from Erdogan's ruling AK party to breach the ban. "I am able to tweet because my DNS settings allow it. That will probably be banned tomorrow as well. I hope that all those who are cursing and using fake accounts will have learnt their lesson," he tweeted, as usual all in capitals.

The first cabinet member to post a tweet after the ban came into effect was the deputy prime minister, Bülent Arinç, who informed his 1.34 million followers of an election rally in the city of Manisa. His message was retweeted more than 1,000 times in the first hour, causing much ridicule:

"Oh dear, be careful, Twitter has been banned by the "national will"," replied Ayse Cavdar, an academic and journalist. "Don't show up here. Otherwise the "national will" will close you down, too."

Meanwhile, the deputy prime minister, Ali Babacan, said he expected the ban to be temporary. "I don't think this will last too long. A mutual solution needs to be found," Babacan told a local TV channel on Friday.

In a rare act of defiance, the Turkish president, Abdullah Gül, openly criticised the ban – via his Twitter account. "The shutdown of an entire social platform is unacceptable," he tweeted. "Besides, as I have said many times before, it is technically impossible to close down communication technologies like Twitter entirely. I hope this measure will not last long."

Later on Friday afternoon several Twitter users in Turkey reported that they were able to access the site again while Turkey's Telecommunications Authority (TIB) denied that the ban had been removed.

Meanwhile Twitter is looking into legal action against the ban and has hired a lawyer to discuss the case with the telecommunications authority in Ankara.Industry minister Fikri Işik said that Twitter needed a permanent legal representative in Turkey to facilitate all negotiations with the Turkish authorities.

Turkey's main opposition Republican People's party (CHP) also announced that it would challenge the blocking of Twitter in court, citing a violation of personal freedoms. The local bar association filed a separate criminal complaint against the ban.

Social media played a big role during last summer's anti-government protest, prompting Erdogan to call Twitter "a menace to society".

Twitter has also been used to disseminate a series of apparently incriminating audio recordings suggesting massive corruption inside the Turkish government.

Turkish officials stressed their discontent about Twitter's "refusal to obey Turkish laws" and to remove individual Twitter accounts that, according to court rulings, breached "privacy rights".

Many expect more explosive revelations to be made via Twitter in the week running up to local elections on 30 March. Two weeks ago Erdogan threatened to ban Facebook and Twitter, accusing social media users of abusing these platforms for a "smear campaign" against his government.

Several human rights groups strongly condemned Turkey's move, warning that the ban spelled a further worrying move towards increased authoritarianism in the country.

"The decision to block Twitter is an unprecedented attack on internet freedom and freedom of expression in Turkey. The draconian measure, brought under Turkey's restrictive internet law, shows the lengths the government is prepared to go to prevent anti-government criticism," said Andrew Gardner, Amnesty International's Turkey researcher, in a public statement.