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U.S. families of terrorism victims have opened up a new legal battle against social media companies, arguing the websites are breaking anti-terrorism laws by letting themselves be used by the Islamic State for recruitment and propaganda.

Two lawsuits, brought by civil rights lawyer Nitsana Darshan-Leitner on behalf of victims’ families of the 2015 Paris and 2016 Brussels attacks, are testing the limits of the U.S. Anti-Terrorism Act by claiming YouTube and Twitter essentially provide material support to the Islamic State by being a conduit for their communications.

Legal analysts said it’s a novel claim, but Ms. Darshan-Leitner argues that the companies need the prodding.

Her lawsuit against Twitter says that the company even attempts to shield terrorists from federal investigators by fighting subpoenas, all while using its anti-harassment policy to shut down the accounts of anti-terrorism activists who reported suspected terrorist-friendly accounts.

“It’s a constitutional issue. It’s a very difficult issue. It’s to balance between freedom of speech to security, which we know is not a simple thing to do,” Ms. Darshan-Leitner said. “The national security value must govern.”

The lawsuits ask the court to grant a financial reward to the families and for Twitter and Google to be found in violation of the Anti-Terrorism Act.

Twitter did not respond to a request for comment.

Google, which runs YouTube, declined to comment, but has filed a motion to dismiss the lawsuit. A court hearing is scheduled for later this month.

Last month Google did announce policy changes on terror-related content, saying it would try to do more to screen videos uploaded to YouTube, and increase the number of people looking to flag inappropriate content.

Stephen Klein, an attorney with the Pillar of Law Institute, said the social media companies are likely protected from liability under the Communications Decency Act, which courts have interpreted to mean that social media websites are conduits, not publishers, of the information that appears on them.

One federal court has already dismissed Ms. Darshan-Leitner’s lawsuit against Facebook, citing the CDA’s protections.

“While superficially content-neutral, this attempt to draw a narrow distinction between policing accounts and policing content must ultimately be rejected,” ordered District Judge Nicholas G. Garaufis.

Jeffrey Addicott, a law professor at St. Mary’s University in Texas, said to make a case under the Anti-Terrorism Act, someone would have to show the companies intended to support terrorists and provided tangible help, such as money, supplies or training.

He said even though the Islamic State and other terrorist organizations use social media to communicate, “they do not endorse of provide the type of material support envisioned by the legislation.”

“This is not to say that a California federal judge may allow the case to go forward, but ultimately the case will fail,” Mr. Addicott said.

Nathan Tarnor, an attorney at Hagens Berman, said Ms. Darshan-Leitner could make the case that the social media companies give terrorist organizations access to sophisticated communications networks, which facilitate acts of terrorism.