The government must do more to promote integration and not look away from problems out of a fear of “being labelled racists”, Kemi Badenoch has told the BBC.

The Conservative leader said the UK had “taken for granted the fact that we have been a fairly cohesive society” but warned “things are fragmenting”.

But she warned that immigration levels were too high, adding that “If people come in too quickly, it’s impossible to integrate”.

She also defended remarks made by Shadow Justice Secretary Robert Jenrick who said Handsworth in Birmingham was “one of the worst-integrated places” he had ever been to.

In a recording reportedly made in March during a dinner and published by the Guardian, the shadow justice secretary said he had not seen “another white face” in the hour and a half he spent in Handsworth filming a video about litter.

He went on: “I want to live in a country where people are properly integrated.

“It’s not about the colour of your skin, or your faith, of course it isn’t. But I want people to be living alongside each other, not parallel lives.”

Asked about the comments, he told the BBC’s Chris Mason: “There are numerous parts of our country now where the same story is happening, and at the extreme levels, a lack of integration leads us into a very dark place as a country.

“We’re here in Manchester today, a week on from a terrible terrorist attack where a man who lived in our country for 30 years clearly wasn’t well integrated, clearly didn’t share British values because he went on to murder British Jews.”

His earlier comments have provoked criticism from opposition politicians.

Badenoch defended his comments, and said they were made during the bin strike in Birmingham.

“Rubbish piled up in the streets and what were the politicians focusing on – airports in Pakistan and Gaza,” she said.

“We need people to have their eyes focused on what is going on in this country.”

Badenoch told the BBC’s political editor Chris Mason that immigration was something her party “got wrong” in government and that was why she was coming up with “tougher” policies including leaving the European Convention on Human Rights and a plan to deport 750,000 illegal immigrants in five years.

In addition to reducing levels of immigration, Badenoch also said the government needed to be “making sure people understand what our values are”.

“Not being tolerant of intolerance, not looking away because we are afraid of being labelled racists.

“That’s one of the things I saw with the grooming gang scandal – a lot could have been done to stop that.

“State authorities were too scared to intervened because they were afraid of looking prejudice.”

While Badenoch was conducting a string of media interviews, Reform UK announced that 20 Conservative councillors were defecting.

The Conservatives are facing a political challenge from Reform, who have been leading in recent opinion polls.

At the Conservative conference this week, Conservatives have been discussing how to respond to the threat posed by Nigel Farage’s party.

A slew of policy announcements during the conference has counteracted the accusation, levelled at Badenoch, that she had previously left a vacuum that Reform had filled.

But Badenoch dismissed the announcement of the defections as “a stunt”.

“Some people just want to go with whoever is winning. Lots of people jump around with the polls,” she said.

“If they are not tough enough to deal with difficulties in opposition, they are not going to be tough enough to deal with difficulties in government.”

She also repeated that she did not want to do a deal with Reform UK.

“How could you explain going into a pact with a party that wants to increase welfare?” she asked.