Newspaper groups fear the government’s proposals to crack down on damaging social media content could inadvertently result in censorship of their own websites.

The Daily Mail is among the outlets that have warned about the government’s proposals to ask Ofcom to ensure British websites do their best to reduce online content that is legal but harmful to society.

The newspaper ran a prominent comment piece on Thursday arguing that the new law “may lead to state censorship”, suggesting the rules could result in its popular sister website MailOnline being regulated and forced to “react to concerns over harmful content”.

As a result industry lobby groups, which have long campaigned for regulation of Facebook and Google, are asking the government to formally commit to a specific opt-out from the online harms law for news publishers. This raises the prospect that a small group of traditional news outlets will be specifically exempted from rules applied to almost every other major website.

Under the proposals announced on Wednesday, British websites that publish user-generated material – including online comments – will be required to produce a code of conduct setting out which material they will allow on their sites. Ofcom will check whether they are living up to their own stated standards.

The other scenario feared by mainstream publishers is that, if the likes of Facebook feel obliged to take down content that is legal but considered harmful to society, social networks could start blocking links to disturbing news stories.

Ofcom could ultimately end up in the position of looking at whether a social network should have removed a link to a particular mainstream article for being harmful.

Resolving the issue will be one of the first challenges for the newly appointed culture secretary, Oliver Dowden, amid signs that Downing Street is concerned about coverage suggesting the new rules could impact on the freedom of the press.

Ministers have repeatedly promised that there will be an exemption for news publishers but on Thursday the News Media Association (NMA), the group which represents almost all traditional British newspaper publishers, including the Guardian, said it was seeking an “explicit exemption on the face of any legislation for news media publishers and their journalism which underpins our democracy”.

The organisation insisted that the legislation should be “designed to crack down on online harms propagated by the tech giants”, arguing newspapers were already responsible for what they publish in the courts and through voluntary systems of press regulation.

The government has long insisted they would provide some opt-out but it is unclear how this would be implemented. If the government chose to exempt only members of the News Media Association in the law then it could end up with a two-tier system. This could see traditional news outlets given a legal opt-out from the legislation while non-NMA members – such as digital-only sites Vice, HuffPost and BuzzFeed News – ending up being caught by the obligation.

A government spokesperson pointed to the commitment by former culture secretary Jeremy Wright to ensure that news organisations were not caught by the new law.