The Church of Scientology does not try to get sites taken down. They do get their copyrighted material which is being illegally posted removed.
From the Church itself:
"The Church has emerged as an experienced, recognized voice in the debate surrounding the survival of both free speech and intellectual property rights in cyberspace.
The Internet offers a wealth of readily accessible knowledge but as with any new frontier, the acts of a lawless few can jeopardize the promise of progress and compromise the rights and freedoms of the responsible, law-abiding majority. The Internet can be abused as easily as it can be employed for good. Finding “free speech” a convenient cloak to disguise their aberrational conduct, denizens of the darker corners of cyberspace espouse perversion, exploit children through molestation and pornography, commit fraud and other forms of commercial crime, invade zones of personal privacy and infringe the intellectual property rights of artists, musicians and thinkers.
The Church of Scientology has stepped into the breach in defense of the free exchange of ideas and in the name of respect for the rights of everyone and has emerged as an experienced voice in the debate surrounding the survival of free speech and intellectual property rights in cyberspace. For example, as a result of a watershed case involving the potential liability of Internet access providers, the Church pioneered a protocol to enable them to respond quickly to Internet abuses without interfering with the privacy or free speech interests of the vast and lawful majority of Internet users.
Likewise, in court decisions in Europe and United States, procedures were developed by the Church to reveal the identities of individuals responsible for transmitting unlawful material such as child pornography and counterfeit copies of intellectual property, without unnecessarily intruding into the privacy of others. The Church’s experiences and accomplishments in this regard have been guideposts for legislators in many countries who are attempting to fashion laws and regulations affecting the Internet.
When reason and law have gone unheeded, the Church has taken legal action to protect Scientology intellectual properties from unauthorized copying and distribution resulting in landmark legal precedents which secure and preserve the freedoms and legal rights of all who travel the information superhighway.
The result of the Church’s dedication has been to the benefit of both freedom and the Internet itself. "
There was an issue of a well-known anti-Scientology site being "shut down by Google" in 2002. Using the DMCA (Digital Millennium Copyright Act) the church requested that Google remove its links to pages which were infringing copyrights. As far as I can tell, any removal of the site was done by its webmaster for publicity. Scientology is only asking the copyright laws to be upheld. They do not file suit or request takedown of every critical site, but they will defend their copyrights. How is that wrong?
The Digital Millennium Copyright Act