This is a great question. While smoking has become very widespread in the early twentieth century, an increase in contracting lung cancer, until then a rather rare disease, was also noticed. Noticing the rise in disease, though, does not mean that people would see what seems to us like an obvious correlation.

Richard Doll and A. Bradford Hill in their article pathbreaking "Smoking an Carcinoma of the Lung" noted many of the obstacles to realizing this correlation. Some only noticed that it hit "older age groups" particularly hard. Others noticed that the "rise seems to have been particularly rapid since the end of the first world war". Many wondered whether it was really an increase or do we just diagnose better now. The two main causes identified before their study, which became known as "The British Doctors Study", were "a general atmospheric pollution from the exhaust fumes of cars" etc and "smoking", but hitherto studies of this second cause were rare and rudimentary. You can find their study here.

Recognition took time. Sure, the Reader's Digest published about the dangers of smoking, detailing specifically the risks of lung cancer and heart disease which were then known, but the first well-publicized announcements made by scientific bodies could only be made around 1957 when a comprehensive literature review of over 7,000 items was released in the United States. It was published in Science, May 1957, 1258 (3258): 1129-1133. A similar report was published in Britain in 1962.

Much of the work here was done in the 1960s under the leadership of Luther Terry as the Surgeon General. These actions included a televised speech warning ordinary Americans, a decision of the Federal Trade Commission to require manufacturers of cigarettes to include warnings on the packaging as well as in advertisements.

He was preceded by some moralist movements, akin to temperance movements, like that of Willem Kieft in the 17th century. Connecticut fined public smokers and, after 1647, those who smoked without a doctor's encouragement. And yet, when General John J. Pershing, somewhat following George Washington, suggested that "tobacco as much as bullets" is needed to win World War I. See Eric Burns, The Smoke of the Gods: A Social History of Tobacco (Philadelphia: Temple University Press, 2007), 101-102, 165.

In China, where medical writings were aware of some detrimental health effects of tobacco, including addiction and, later, issues with women's fertility, it took a lot of time to have that penny drop. It was only in the 1990s that anti-tobacco activism became serious, and some Chinese still believe that "one or two cigarettes a day can prevent malaria" and that "puffing on a cigarette" can "reduce bloating of the stomach" or "cure food poisoning". See Carol Benedict, Golden-Silk Smoke: A History of Tobacco in China, 1550-2010 (Berkeley: University of California Press, 2011), 252.

As you can see, different societies had varying trajectories. in Russia, to give one last example, Tsar Mikhail Feodorovich banned tobacco in 1627, and we are not sure why. Probably a combination of ecclesiastical disapproval of foreign influence and the debts of military officers who spent too much on tobacco. But tobacco made a come-back, and then was fought against by temperance activists, and only then, in the Soviet Union, for health reasons too. But this was not immediate: In the same Soviet Union, my grandpa, like many other kids (as Catriona Kelly shows in an article), smoking was an essential part of becoming a man. This was in 1953 for my grandpa (who was born in 1941, so he was 12) -- for others, it remained even when it was clearly related to cancer.

I hope this helps!