did the U.S, England, and France do anything during World War II to prevent the extermination of European Jews?
The US, France, England, and the Soviet Union did very little to stop the Holocaust, but not for the reasons some may think.
There were some efforts to save Jews, mainly by underground organizations, like the Marquis in France; and by a few individuals like Schindler. But, there were few concerted efforts by the government and forces of the USA, Britain, France, or the Soviet Union.
France, England, and the Soviet Republics (Russia, Ukraine etc.) had long histories of anti-semitism. Continuation of this anti-semitism would naturally lead to apathy towards the Holocaust; but this didn't apply to the USA's generally more tolerant society, and it wasn't the main reason for European lack of action as well.
The main reason was nobody knew. We didn't have social media, or TV. Radio was the only newsmedia faster than the Newspaper. Nazi Germany was very secretive about its anti-semitic policies, and kept most of the concentration camps in secluded areas that were hard to find, and off-limits to most journalists. Since the reports that did get out were so unbelievable, and so few, most thought they were hoaxes. It wasn't until Spring 1945 when Allied forces in Southern Germany and Poland began discovering and liberating the concentration camps, that the full scope and scale of the Holocaust was discovered.
After the camps were discovered, Allied forces began a concerted effort to find and liberate all of the camps, and get humanitarian aid to these camps.