I found a few somewhat related question but I am looking for specific 'aha' moment or study that clearly shows this.
I envision this graph with a before we knew of hand washing and after and it will look something like
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(where x is time and y is the number of sick people)
Is this true is any way? How did people teach this? Was there an uproar? Maybe people were offended thinking they are clean enough?
This isn't a question about history. Did you mean to post in in /r/AskScience, perhaps?
This might be the study you're looking for: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1633559/ it includes graphics just as you describe, see figure 1 and 2.
It measures deaths after child birth and infection was a major cause of maternal deaths. It wasn't until better hygiene was deployed that this cause of death plummeted. It took decades to convince the medici.
Around 1850 dr. Ignaz Semmelweis showed solidly how infections decreased if his students washed their hands (with carbolic soap) when they went from dissecting a corpse to delivering a baby. He was ridiculed. The establisment could not entertain the idea that they themselves could be responsible for so many deaths.
(A 100 years earlier another doctor, Alexander Gordon, had wondered about the same transmission of disease and he was run out of town.)
In 1879 Pasteur showed proof that bacteria were present in the infections that lead to young mothers dying. "Anti-septics" became a thing, meaning to kill bacteria, and a scientific search was on for stuffs that kill bacteria. In its wake followed a growing interest in "a-septics" meaning preventively keeping bacteria away from open wounds. This now lead to hand washing.
But it wasn't untill 1929 that the British College of Obstetricians recommended that at every delivery, masks and rubber gloves, which reached above the elbows, should be worn. In 1934 the maternal death toll due to infections plummeted (fig 2)