This is complicated. It depends. The media covered Nazi persecution and massacres of Jews from the time the Nazis gained power in 1933. How well it was covered depended on the publication. Jewish periodicals were obviously more thorough, but not uniformly so. For example, some communist and socialist Jewish newspapers in America were more concerned about ideological battles than what was happening to Jews in Europe. News then, as now, was dependant on new, terrible things happening for it to make the front page. Thus, events such as the Night of Broken Glass were widely covered, but the day to day persecution, starving people, stranded refugees, etc. didn't make a lot of news.
When what appeared to be incredibly outlandish reports started emerging, such as the massacre of tens or hundreds of thousands of Jews, many publications expressed disbelief and wouldn't run the story, might bury it in the back, or might note that the only source was from a Jew (the implication being that it wasn't as trustworthy). This stemmed from the fear of being accused of printing "atrocity stories," which came into vogue in the First World War when media reported "fake news" of Germans or whoever doing such unspeakable things that turned out to have never happened, but had been propaganda. There was also a stereotype against Jews that they tended to exaggerate.
It was also frequently stressed that Jews were but one group among several who were in a bad way. Protestants, Catholics, and socialists were the others most often cited. Yes, even after it was reported that millions of Jews were dead, the media didn't recognize a Holocaust or genocide like we do today. As far as leaders recognizing the situation for Jews in the media, this happened rarely and only towards the end of the war.
Some books to check out on this are Beyond Belief by Deborah Lipstadt and Buried by the Times by Laurel Leff.
It was known around 1942. Even published.
Some very very brave men managed to flee KZs and smuggle out information.
One known "official"report was around 1942 done by Jan Karski, supported and Organized by the polish Home army and resistance.
Karski was able to get into the Warsaw ghetto and documented the horrible Situation there.
He also managed to get into a transit lager that funneled the trains towards Belzec and flee successfully.
He managed to smuggle a microfilm out as well. All that information was collected into a Note, officially distributed by a polish politician in exile, Edward Raczyński.
The name of the Report:
"The mass extermination of Jews in German occupied Poland"
it was addressed to the united nations and the allied powers.
Karski met other important leader to get the information around,
for example he met roosevelt 43.
Some people had troubles believing it.
If you want to read the report, here are images of it:
http://www.projectinposterum.org/docs/mass_extermination.htm
I think he also got (understandable) rather bitter- he told he Truth he obtained at great risk of his own life, he did so much to make it widely known, to get the the powers to act.
We know there are aerial photos of major KZs, Powers knew where some of them were.
Still they weren't prioritizes, not targeted bombing of railway tracks leading to them, for example.
He was of the opinion, it was a deliberate choice, maybe out of pragmatism, but still a choice.
Karski said in an interview:
"The Allies considered it impossible and too costly to rescue the Jews, because they didn't do it. The Jews were abandoned by all governments, church hierarchies and societies[...]Now, every government and church says,
"We tried to help the Jews", because they are ashamed, they want to keep their reputations. They didn't help, because six million Jews perished, but those in the government, in the churches they survived. No one did enough."
(http://remember.org/educate/hrintrvu)
Besides Karski's, other reports were made. Some published after the end of the war, but still the knowledge was around- not necessarily as precise, but people knew hat the Jews were getting exterminated.
There are the Auschwitz reports- a collections of reports, one obtained from men managed to flee a daring escape from Auschwitz.
(They masqueraded as SS officers and just drove out of the Lager. It was a "right under the enemies eyes" situation^1)
the Vrba–Wetzler report,(Rudolf Vrba and Alfréd Wetzler) who managed to escape Auschwitz on the 10.04.1944 and protocol their experiences. It also contains reports and experiences of other inmates, so its more protocols.
The Mass murder was known, so they tried to estimate numbers, had a sketched approximated plan about the factories/industries in auschwitz (Ig Farben, Buna, Krupp, Siemens) and also about the Layout of gas chambers, obtained by members of the Sondereinsatzkommando, one Fillip Müller, a slovak, one of the few SK who survived.
(The report contains some irregularities, but seeing that its witness by people who only survive-we know that Trauma has an effect on memory, so irregularities do not mean intentional lies, it means often that the situation was so stressful and excruciation, the human self defended itself by forgetting things or misremembering.^2
This Report is credited with changing the position of Horthy, who stopped deportations towards Auschwitz from hungary (12k/day)
The allies knew since 1942, here an article in the NYTimes for november 1942.
(full articles are gated, but the excerpt says enough)
https://www.nytimes.com/1942/11/25/archives/himmler-program-kills-polish-jews-slaughter-of-250000-in-plan-to.html and:
https://www.nytimes.com/1942/11/25/archives/details-reaching-palestine.html