(Adds Maduro and Obama quotes, details on Puerto Ordaz and Caracas)
By Tomas Sarmiento and Diego Ore
CARACAS, Feb 19 (Reuters) - A local beauty queen died of a gunshot wound on Wednesday, the fifth fatality from Venezuela's political unrest, as imprisoned protest leader Leopoldo Lopez urged supporters to keep fighting for the departure of the socialist government.
Tensions have risen in Venezuela since Lopez, a 42-year-old Harvard-educated economist, turned himself in to troops on Tuesday after spearheading three weeks of often rowdy protests against President Nicolas Maduro's government.
The latest victims of the unrest included college student and model Genesis Carmona, 22, who was shot in the head at a protest on Tuesday in the central city of Valencia. She died later in a clinic.
"How long are we going to live like this? How long do we have to tolerate this pressure, with them killing us?" a relative, who asked not to be named, told Reuters.
"She only needed one more semester to graduate," he added of Carmona, who had been studying tourism and had won the 2013 Miss Tourism competition in her state.
Three people were shot dead in Caracas after an opposition rally a week ago, and a fourth person died after being run over by a car during a demonstration in the coastal town of Carupano. Scores of people have been arrested or injured.
State television channel VTV said the mother of one its employees died while being rushed to hospital in Caracas. VTV said she suffered a heart attack while the ambulance carrying her was stuck in gridlock due to opposition supporters blocking roads.
"We cannot underestimate those fascist groups whose boss is behind bars," Maduro said in a nationally televised speech late on Wednesday. "I'm not playing with democracy. I do not accept that they challenge the Venezuelan people and our Constitution."
Speaking in Mexico, U.S. President Barack Obama criticized Maduro's government for arresting protesters and urged it to focus on the "legitimate grievances" of its people.
Instead of "making up false accusations" against three U.S. diplomats expelled this week, he said, Venezuela's government should release detained protesters and engage in a real dialogue. "All parties have an obligation to work together," Obama said.
Lopez has urged his supporters to keep fighting for the departure of Maduro's socialist administration.
"Today more than ever, our cause has to be the exit of this government," he said, sitting by his wife in a pre-recorded video that was to be released in the event he was jailed. (http://t.co/uJGiXVm0AV) "Let's fight. I will be doing so."
There were sporadic clashes across Venezuela on Wednesday. Rival groups scuffled outside the Caracas court where Lopez was due, while student demonstrators also blocked several roads in the capital with burning trash.
In western Tachira state, security forces and protesters fought in the streets for about two hours. Two students were injured, vehicles were damaged or destroyed, and local monuments were charred, witnesses said.
In southern Puerto Ordaz city, pro- and anti-government marchers clashed in the street, witnesses said, with police firing teargas to quell the fighting.
Three government supporters suffered gunshot injuries and the two sides faced off with sticks and stones, the witnesses said. Maduro said one of the injured government supporters was in a grave condition.
Police fired teargas to disperse protesters from a square in the wealthier east of Caracas. Sporadic shots echoed in the surrounding streets, and bands of motorcyclists roamed the area.
The demonstrators are calling for Maduro's resignation over issues ranging from inflation and violent crime to corruption and shortages of goods.
Maduro, who was narrowly elected last year to replace Hugo Chavez after his death from cancer, says Lopez and others in league with the U.S. government are seeking a coup.
Street protests were the backdrop to a short-lived ouster of Chavez for 36 hours in 2002, before military loyalists and supporters helped bring him back.
Though tens of thousands joined Lopez on the streets when he turned himself in on Tuesday, the protests have so far been much smaller than the wave of demonstrations a decade ago.
There is no evidence the military, which was the decisive factor in the 2002 overthrow, may turn on Maduro now.
Lopez was being held at the Ramo Verde jail in Caracas. His first court appearance was expected to take place there.
In an intriguing twist to the drama, Maduro said the powerful Congress head Diosdado Cabello, seen by many Venezuelans as a potential rival to the president, personally negotiated Lopez's surrender via Lopez's parents.
Cabello even helped drive him to custody in his own car given the risks to Lopez's life from extremists, Maduro said.
With local TV providing minimal live coverage of the street unrest, Twitter, Instagram and Facebook have become the go-to media for many Venezuelans desperate for information.
However, many social media users are indiscriminately tweeting images without confirming their origin, leading to manipulation and gaffes including footage of unrest in Egypt and Chile being passed off as repression in Venezuela.
Detractors call Lopez a dangerous and self-serving hothead. He has frequently squabbled with fellow opposition leaders, and was involved in the 2002 coup, even helping arrest a minister.
Though the majority of demonstrators have been peaceful, a radical fringe has been attacking police, blocking roads and vandalizing buildings. Rights groups say the police response has been excessive, and some detainees say they were tortured. (Additional reporting by Andrew Cawthorne and Diego Ore in Caracas; Javier Farias in Tachira; German Dam in Puerto Ordaz and Mark Felsenthal in Toluca, Mexico; Writing by Andrew Cawthorne; Editing by Chizu Nomiyama, Ken Wills and Mohammad Zargham)