At least seven people were killed in the bombing, which inflicted huge damage on government buildings in Oslo city centre.
Police said at least nine more died at the camp, on an island outside Oslo.
One witness later said he had seen more than 20 bodies on the island, but police have not confirmed this.
Prime Minister Jens Stoltenberg, whose Oslo offices were among those damaged by the bomb, described the situation as "very serious".
Norwegian media reports said the shootings on the island, on the Tyrifjorden lake, were carried out by a man in police uniform.
Police said the suspected gunman had been arrested, and later that he was also linked with the bomb attack. Reports described him as tall and blond.
No group has said it carried out the attacks.
Foreign Minister Jonas Gahr Store told the BBC the country was in "deep, deep shock".
"Norway is today dealing with a double attack on its democracy, on its government buildings and on its finest youth wanting to engage in politics."
Hours after the bomb struck Oslo, officials said some people were still inside the damaged buildings, some of which were on fire.
Television footage from the government quarter showed rubble and glass from shattered windows in the streets and smoke from the fires drifting across the city. The wreckage of at least one car could be seen.
All roads into the city centre have been closed, said national broadcaster NRK, as security officials evacuated people from the area, fearing another blast.
Mr Stoltenberg, in a telephone call to Norwegian television, said all government ministers were safe.
He said he had been advised by police not to reveal his current location, but he is not thought to have been in central Oslo on Friday.
"Even if one is well prepared, it is always rather dramatic when something like this happens," he said.
Egil Vrekke, Assistant Chief Constable of Oslo police told the BBC the rescue operation in Oslo was ongoing.
"We are issuing warnings just [to] make sure people are not in the area in case there are further explosions," he told the BBC.
"We have cordoned off large areas. There are bomb experts at the scene investigating whether there are other devices in the area."
A spokesman for Oslo University hospital said 10 people had been taken there for treatment, some with serious injuries.
A few hours after the explosion, the gunman opened fire at a camp in Utoeya for young members of the Labour Party.
Andre Scheie, who said he went to the island after the shooting to help evacuate people, told reporters he had seen at least 20 bodies.
NRK journalist Ole Torp said there were reports the gunman had been armed with a handgun, an automatic weapon and a shotgun.
"He travelled on the ferry boat from the mainland over to that little inland island posing as a police officer, saying he was there to do research in connection with the bomb blasts," he told the BBC.
"He asked people to gather round and then he started shooting, so these young people fled into the bushes and woods and some even swam off the island to get to safety."
Mr Stoltenberg had been due to visit the camp on Saturday. He told TV2 the situation there was critical.
Police have not yet commented on the number of victims, but police spokesman Anders Frydenberg told the BBC that soldiers were now on the island.
Foreign Minister Mr Store, who visited the camp on Thursday, praised those who were attending.
"The country has no finer youth than young people who go for a summer camp doing politics, doing discussions, doing trainnig, doing football, and then they experience this absolutely horrendous act of violence."
'Focus on rescue'
State Secretary Kristian Amundsen said Friday was a public holiday in Norway so the government offices were not as busy as they might usually have been.
"But there are many hundreds of people in these buildings every day," he told the BBC.
Journalist Hanne Taalesen on island attack: "There are reports that youths hid in bushes"
"We have to focus on the rescue operation - there are still people in the building, there are still people in the hospital."
Reuters said the oil ministry was among the other government buildings hit, while NRK journalist Ingunn Andersen said the headquarters of tabloid newspaper VG were also damaged.
"It's complete chaos here. The windows are blown out in all the buildings close by," she told AP.
Oistein Mjarum, head of communications for the Norwegian Red Cross, which has offices nearby, said the blast could be heard across Oslo.
"This is a very busy area on Friday afternoon and there were a lot of people in the streets, and many people working in these buildings that are now burning," he said.
Mr Mjarum said people across Oslo and Norway were in shock.
"We have never had a terrorist attack like this in Norway - if that's what it is - but of course this has been a great fear for all Norwegians when they have seen what has been happening around the world."
The United States has condemned the "despicable acts of violence" in Oslo, while the President of the European Council, Herman Van Rompuy, said he was "deeply shocked" by "these acts of cowardice for which there is no justification".
Analysis
Gordon Corera Security correspondent, BBC News
Police are saying that they believe the car bomb and the shooting are linked and they are thought to have a suspect in custody from Utoeya. That means they should be getting a clearer idea of who was behind this attack.
An al-Qaeda linked group is still a strong possibility, based on opposition to the role of Norwegian forces or issues linked to the publication of cartoons of the prophet. There were arrests last year in Norway linked to an international terrorist plot.
However, at this stage other possibilities, including domestic extremists, cannot be ruled out.
Constructing a large car bomb requires a degree of sophistication and the crucial factor for the police will be establishing how many people are behind this attack, whether any are still at large and to whom they might connected.
http://www.bbc.co.uk/news/world-europe-14256712
