The typhoon also triggered flooding and landslides in the nearby Philippines, where authorities said four people were killed south of the capital Manila.
Gaemi, with sustained wind speeds of 190 kilometres (118 miles) per hour, was initially expected to hit northeast Taiwan by 10 pm (1400 GMT).
However, a forecaster with Taiwan's Central Weather Administration said the typhoon was currently 30 kilometres (18 miles) from the eastern coast of Hualien and could make landfall "in the next few hours".
"Gaemi is expected to be the strongest typhoon in eight years to make landfall in Taiwan since Typhoon Nepartak," forecaster Huang En-hong told AFP, referring to a 2016 superstorm that killed three and injured hundreds.
The typhoon had caused torrential rain and whipped up winds across Taiwan even before its arrival, killing one motorist in southern Kaohsiung city who was crushed by a falling tree, authorities said.
Nearly 60 people had been injured by Wednesday afternoon, authorities said.
President Lai Ching-te, who took part in a morning emergency briefing, urged the public to "put safety first" during what would be the first typhoon to make landfall in Taiwan this year.
"I hope that through our joint efforts, impact from the typhoon can be minimised... I also encourage fellow citizens across the country not to go out unless necessary during the typhoon, especially not to dangerous places," Lai said.
The weather forced the self-ruled island to cancel some of its annual Han Kuang war games -- which test preparedness for a Chinese invasion -- but an anti-landing drill went ahead as scheduled on Wednesday morning on Penghu island, west of Taiwan.
Authorities evacuated more than 4,000 people living in precarious conditions in the northern regions, particularly Hualien -- a mountainous area with a high risk of landslides.
Trains and ferry services were suspended and hundreds of international and domestic flights were cancelled on Wednesday.
"We expect that the impact of the typhoon will be extended to four days (until Friday)," said Taiwan's Central Weather Administration chief Cheng Jia-ping, adding that the public would need to "take precautions against heavy rain and strong wind".
Gaemi was expected to pass through the Taiwan Strait and hit China by Thursday in eastern Zhejiang and Fujian provinces, where authorities have issued a red storm alert.
- 'Exercise strong vigilance' -
Gaemi swept past the Philippines, exacerbating seasonal monsoons and triggering widespread flooding in Manila that turned streets into rivers and trapped vehicles.
Landslides killed four in Batangas province, south of the capital, and blocked three major roads in mountainous Benguet province, police and disaster officials said.
Weather authorities in Japan's southern island region of Okinawa urged residents to "exercise strong vigilance" against storms, high waves and floods.
Massive waves crashed ashore in Taiwan's northeastern Yilan county, while market vendors worked quickly to protect their stalls with canvas and shops had taped-up windows.
A fisher surnamed Hsu tied down his boat at a typhoon shelter in a harbour crowded with docked vessels.
"I am worried about the typhoon -- the boats are my tool for making money," he told AFP.
Government offices were closed and streets emptied in the capital Taipei, while some stores had their entrances sandbagged to prevent potential floodwater.
Taiwan is accustomed to frequent tropical storms from July to October but experts say climate change has increased their intensity, leading to heavy rains, flash floods and strong gusts.
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