The truck, which is said to have entered Britain via the Welsh port of Holyhead on Saturday, was found by ambulance workers at Waterglade Industrial Park in Grays, a town just by the River Thames around 25 miles (40 kms) east of central London.
"This is a tragic incident where a large number of people have lost their lives. Our enquiries are ongoing to establish what has happened," Essex Police Chief Superintendent Andrew Mariner said.
"We are in the process of identifying the victims, however I anticipate that this could be a lengthy process."
A 25-year-old-man from Northern Ireland who was driving the truck has been arrested on suspicion of murder. He remains in custody. He has not been charged or identified.
A cordon has been put in place and access to and from the industrial park remains closed.
"We are working with Thurrock Council (the local authority) to mitigate against any impact our investigation scene will have locally." Mariner said.
Prime Minister Boris Johnson pledged in a tweet to work closely with Essex Police to establish exactly what happened.
"My thoughts are with all those who lost their lives & their loved ones," he said.
Police have not formally linked the deaths to people trafficking but a link is assumed because of the way the victims were crammed into the truck container.
The tragedy recalls the death of 58 migrants in 2000 in a truck in Dover and the deaths in 2015 of 71 migrants from Syria, Iraq and Afghanistan found suffocated in the back of a refrigerated truck abandoned in the emergency lane of a highway near Parndorf, Austria, close to the Hungarian border.
Smaller numbers of migrants have occasionally been found dead in trucks in Britain in recent years.
Bulgarian authorities said they could not yet confirm that the truck had started its journey from Bulgaria.
"We are in contact with our embassy in London and with British authorities," foreign ministry spokeswoman Tsvetana Krasteva said.
Jackie Doyle-Price, who represents Thurrock in parliament, said in a tweet that "people trafficking is a vile and dangerous purpose ... Let's hope they bring these murderers to justice."
Essex police say the force has not yet identified the 39 victims or where they came from.
Deputy police chief Pippa Mills said identification "could be a lengthy process" and that a key line of inquiry will be how the truck entered Ireland. It is believed to have traveled from Ireland to Holyhead in Wales on Saturday via ferry.
She refused to describe the gender of the victims.
Seamus Leheny, Northern Ireland policy manager for the Freight Transport Association, said the route that had been used was "unorthodox" since it apparently involved traveling to Ireland and then entering Britain via a ferry to a major passenger port in Wales.
He said that choice may have been influenced by increased security and checks in the major English port of Dover and the French port of Calais.
"People have been saying that security and checks have been increased at places like Dover and Calais, so it might be seen as an easier way to get in by going from Cherbourg or Roscoff, over to Rosslare, then up the road to Dublin," he said.
"It's a long way around and it'll add an extra day to the journey."
Richard Burnett, chief of the Road Haulage Association, said the truck may have traveled from the French port of Cherbourg to the Irish port of Rosslare before continuing by road to Dublin and taking a ferry to Holyhead in Wales.
It would have then traveled by road to Grays where the bodies were found.
He said it is "highly unlikely" the vehicle would have been subjected to a physical check if it had taken that route because those ports have far fewer checks then Dover and Calais, where there are sniffer dogs, monitors, and more advanced technological surveillance.