Thousands came to the street where the tiny girl was decapitated.

They carried Winnie the Pooh stuffed toys, flowers, balloons, cards and even a red bike.

They wrote to her family, with fervent offers of compassion and aid.

The massive outpouring of grief from the Taiwanese over Monday's random killing of the four-year-old, nicknamed Little Light Bulb for her sunny disposition, overwhelmed her mother Claire Wang with gratitude.

But the 44-year-old housewife called on the well-wishers to redirect their efforts to those in need.

"There are many others whose lives are not so wonderful. They are lacking in psychological well-being, lacking in material needs," she said. "Please focus your kind intentions and offers towards making society a better place."

Ms Wang's call, penned on her Facebook page on Friday, underscored what many say is an urgent need for Taiwan to tackle mental health issues and, beyond that, broader concerns about its economy and education.

Little Light Bulb's suspected killer Wang Ching-yu, who is 33 and jobless, has a history of mental illness and drug abuse.

In 2006, he was convicted of drug-related offences but was released. Two years ago, he sought psychiatric treatment at a hospital.

However, he did not have a government-issued disability card, which would have entitled him to health benefits and kept him on the watch list of the authorities.

At his home later, police officers found scribbling on his notepad which read "Killing Sichuan girls can extend ancestral bloodline".

That was apparently his mission on Monday. That morning, he bought a kitchen cleaver at a market and took the train to Neihu district in eastern Taipei.

There, he waited.

At 11am, he spotted and accosted Little Light Bulb and her mother, who were on their way to meet family members. For some reason, he believed the girl was from the mainland Chinese province.

Little Light Bulb was struggling to wheel her strider bike out of a pothole and her mother, who was pushing an empty stroller, thought he was helping her. Instead, Wang reportedly grabbed the child, slashed at her neck and severed her head.

Cases like Wang's, where mental instability apparently played a role, are not isolated in Taiwan any more.

Less than two years ago, university student Cheng Chieh, then 21, embarked on a stabbing spree on the Taipei subway, killing four and injuring 24. He had fantasised about "doing something big" since childhood, he told police. Last May, an eight-year-old died after her throat was slashed in the toilet of her school. The culprit, Kung Chung-an, claimed voices in his head had urged him to do so.

With the latest case, politicians have admitted that Taiwan has been found wanting, both in caring for the mentally unstable and in protecting society from their actions.

Taipei Mayor Ko Wen-je, who is trained as a doctor, told reporters after the incident that Taiwan's "safety net has holes" and that both "preventive and protective mechanisms" will have to be reviewed.

Incoming President Tsai Ing-wen vowed to "double down on efforts to resolve issues of drugs, child safety, police actions (and) psychological and mental health treatment".

Experts say the recent spate of incidents is not a case of there being more severely mentally ill people in Taiwan.

Dr Lee Chau-shoun, a psychiatrist at Mackay Memorial Hospital, says such senseless killings would likely be driven by serious disorders such as schizophrenia and bipolar disorder. These are endogenous diseases caused mainly by genetic and biological factors, so there has been no sharp increase over time. About 70,000 people - or 0.3 per cent of Taiwan's population of 23 million - suffer from them. What has changed though, say those interviewed, is how Taiwan handles mental health issues.

Since the 1995 establishment of a national health insurance scheme covering medical fees, doctors are moving into more profitable specialities like cosmetic surgery, while budgets for less glamorous fields such as psychiatry are shrinking.

Dr Lee's department is in charge of the psychiatric community care centre in New Taipei, which has a population of four million, but it has a budget of NT$1 million (S$42,000) for work that requires five times that amount.

"You may imagine the quality of the final results and the hard work required of the staff," he says.

Strapped of resources and manpower, many psychiatrists have had to make the decision to let "some difficult patients remain in the community and be treated intermittently". At the same time, counselling - a long-term regimen - is not covered by insurance, says Dr Chan I-wen of the Community Services Centre in Taipei.

Taiwanese society remains hazy about mental illnesses. The education system has churned out people armed with cognitive skills but little understanding of their inner well-being and the need to seek help if unwell, says Dr Chan.

She blames the proliferation of virtual gaming for eating into human interaction. Meanwhile, parents work long hours while their children study even longer hours.

She says: "We need a government with leadership to solve Taiwan's economic problems and allow parents to have time to spend with children, while also having the resources to help them."

There is also stigma attached to mental illnesses. Some may not wish to register for the mental disability card, as Wang did not, because of the negative associations.

Acknowledging this, Mayor Ko told reporters: "If society is full of harmony, the mentally ill - even if they are hallucinating - will move in a positive fashion. But when society is full of violence and negativity, they will feel persecuted and will take action to protect themselves."

As Dr Lee puts it: "The killer of Little Light Bulb - if indeed he is mentally ill - is a victim too."