"There should be a U.N. human rights office everywhere," Turk said, stressing that "all states can and should do better on human rights."
The body's rights chief said he had been lobbying U.N. member states to double his office's funding and allow it to broaden its footprint, as it strives to address rights abuses around the world.
"My office is now present in more places than ever. We have gone from just two field presences when we started to 94 presences around the world today," he told journalists in Geneva.
Turk said that his office currently has an annual budget of about $390 million, of which around a third comes from the regular U.N. budget - and the rest is from voluntary contributions.
"We want to see a doubling, which would be about $800 million per year for the organization," he said.
"A strong U.N. human rights office and a healthy, well-resourced human rights ecosystem are of global interest."
Turk pointed to an ever-expanding number of requests from countries for assistance from his office, especially through the U.N. Human Rights Council, asking for its expertise and to support an array of investigations into rights abuses around the world.
The office, established 30 years ago, "has contributed to greater recognition of the centrality of human rights in making and sustaining peace, preventing and halting violations, in fostering accountability," Turk said.
He pointed to how the U.N. rights office works with countries, national rights bodies and civil society, "serving as a bridge" between them.
"We call out violations and set off alarm bells when attacks on, neglect of, or disdain for human rights could set off crises," he said.
"We provide a reality check."
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