I think newspapers vastly overestimate their own power. Let's not forget that 70% of the newspapers in this country went all out for 3 weeks earlier this year to FORCE the electorate to vote Conservative with laughable results (and indeed laughable "reporting").
This, along with the fact that less and less people are reading papers every year (why pay for news that is, at worst total lies and at best horrifically slanted opinion pieces/churnalism) means they'll have less and less power in the future. There will come a time soon when MPs will easily be able to ignore whatever crap papers decide to use as a dog whistle for easily enraged idiots (immigration anyone?) and actually get on with making decent, evidence based, policy decisions rather than having to pander to Rupert Murdoch and Paul Dacre's twisted idea of what's right and wrong.