When the Taliban sent a gunman to shoot Malala Yousafzai last October as she rode home on a bus after school, they made clear their intention: to silence the teenager and kill off her campaign for girls' education.

Nine months and countless surgical interventions later, she stood up at the United Nations on her 16th birthday on Friday to deliver a defiant riposte. "They thought that the bullet would silence us. But they failed," she said.

As 16th birthdays go, it was among the more unusual. Instead of blowing out candles on a cake, Malala sat in one of the United Nation's main council chambers in the central seat usually reserved for world leaders.

She listened quietly as Ban Ki-moon, the UN secretary-general, described her as "our hero, our champion"; and as the former British prime minister and now UN education envoy, Gordon Brown, uttered what he called "the words the Taliban never wanted her to hear: happy 16th birthday, Malala".

The event, dubbed Malala Day, was the culmination of an extraordinary four years for the girl from Mingora, in the troubled Swat valley of Pakistan. She was thrust into the public glare after she wrote a pseudonymous but later celebrated blog for the BBC Urdu service describing her experiences struggling to get an education under the rising power of Taliban militants.

By 11 she was showing exceptional determination, calling personally on the US special representative to Pakistan, Richard Holbrooke, to use his influence to combat the Taliban's drive against education for girls. By 14, she was on the radar of Archbishop Desmond Tutu, who put her forward for the international children's peace prize, and by 15 she became the youngest Nobel peace prize nominee in history.

But such dizzying global attention came at a price. Death threats followed her growing recognition, and on 9 October 2012, following a meeting of Pakistani Taliban leaders, the gunman was dispatched to remove what they called the "symbol of infidels and obscenity".

Multiple operations in Pakistan and the UK followed the attack on the bus, including the fitting of a titanium plate on her left forehead, and a cochlear implant to restore her hearing. She now lives with her family in Birmingham and does what the Taliban tried to stop her doing: goes to school every day. "I am not against anyone," she said in the UN chamber, having taken this day out from the classroom. "Neither am I here to speak in terms of personal revenge against the Taliban or any other terrorist group."

Malala responded to the violence of the Taliban with her own countervailing force: words against bullets. "I do not even hate the Talib who shot me. Even if there is a gun in my hand and he stands in front of me, I would not shoot him."

She spoke confidently, with only an injured eye and a slightly drooping left side of her face to hint at such fresh traumas. There was one other unstated allusion to the horror of her past: she wore a white shawl belonging to a woman who was also targeted by extremists but who, unlike Malala, did not survive to tell the tale: Benazir Bhutto.

"The extremists are afraid of books and pens," the teenager continued. "The power of education frightens them. They are afraid of women. The power of the voice of women frightens them."

She cited last month's attack on a hospital in Quetta, capital of Baluchistan, and killings of female teachers in Khyber Pakhtunkhwa. "That is why they are blasting schools every day – because they were and they are afraid of change, afraid of the equality that we will bring to our society."

And she gave her own opposing interpretation of Islam to the Taliban's. "They think that God is a tiny, little conservative being who would send girls to the hell just because of going to school. The terrorists are misusing the name of Islam and Pashtun society for their own personal benefits. Islam is a religion of peace, humanity and brotherhood. Islam says that it is not only each child's right to get education, rather it is their duty and responsibility."

Such ability to articulate what normally remains unarticulated – to give voice to young people normally silenced – has generated its own response. The "stand with Malala" petition, calling for education for the 57m children around the world who do not go to school, has attracted more than 4m signatures – more than a million having been added in the past few days.

At the start of her speech, Malala said: "I don't know where to begin my speech. I don't know what people would be expecting me to say."

She need not have worried.