By Gram Slattery

SANTIAGO (Reuters) - Chile's students were back on the streets on Thursday, four years after their massive protests won wide public support, this time taking on a left-leaning president whose promise of university reform has fallen short of their demands.

After President Michelle Bachelet announced major legislation in May to make university free for hundreds of thousands of Chileans, tens of thousands of protestors marched through Santiago's broad avenues asking for quite a bit more.

They want students and professors to have more say in how universities are run and for primary schools to be federally controlled - a significant departure from Bachelet's proposals.

The student cause has also broadened, with recent protests folding in demands such as scrapping the nation's dictatorship-era constitution and repealing a law which privatizes marine resources.

The return of the demonstrations has raised questions about how the government of one of Latin America's most developed economies deals with a potent political bloc that keeps moving leftward and enjoys far broader support than any traditional party.

"It seems to me that new reforms need to change the structure of the government itself," said Valentina Saavedra, the president of the University of Chile's powerful student union.

"We have been fighting as students, workers, teachers, because traditional politics aren't working."

Joined by professors and high school pupils, students on Thursday chanted slogans, beat drums, and waved festive banners as they walked along the Alameda, Santiago's main artery. As with many Chilean protests, there were violent incidents as protesters threw molotov cocktails at police and set barricades on fire.

It was perhaps more conventional than other recent protests, such as a naked demonstration before the nation's presidential palace and a mass punting of soccer balls in reference to the regional Copa America soccer championship, which Chile is currently hosting.

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The 2011 protests - the so-called "Chilean Winter" - dealt a significant blow to the popularity of conservative President Sebastian Pinera a year into his mandate and his approval rating never fully recovered. Bachelet, hoping to return to the presidency, made hefty campaign pledges for reform in the wake of those demonstrations.

STUDENTS SHIFT LEFT

Bachelet's approval rating - like Pinera's post-2011 - is mired in the twenties, having plunged from 54 percent when she returned to power for a second term in March 2014.

As well as disillusionment with the speed of reforms, she has been hurt by a weak economy and a string of corruption scandals that have left Chileans angry at the entire political class.

Although Chile has made great strides in reducing poverty in recent decades, like other countries in the region it still suffers from deep inequalities.

That means the students' anti-establishment rhetoric strikes a chord.

A year ago, Bachelet seemed to have brought the protesters into the fold. A clutch of former student leaders, including Camila Vallejo, who earned global headlines for heading the student movement four years ago, are now lawmakers in the coalition government.

But the student leadership has a new face, and has radicalized since 2011. The new generation has distanced itself from the student leaders of old.

"We're realizing that we were tricked by people like Vallejo, because clearly she became famous, the world knows her, and instead of advocating for us in Congress, she turned her back," said Camila Riquelme, a student outside the protestor-occupied University of Chile earlier this week.

Some locals are tiring of the student protests, and analysts say that support for their movement has likely dropped from a poll last September which showed almost 80 percent of Chileans backing the marches.

But with no party garnering support above 29 percent, people still sympathize far more with the students than with traditional politics.

"You have government after government reach some agreement with the students, and the next generation comes along and says, 'Hey, we don't accept this,'" said Robert Funk, a political scientist at the University of Chile.

"The compromises of the past never seem good enough."

(Editing by Rosalba O'Brien, Mary Milliken and Andrew Hay)