Do infrastructure projects like the ones by FDR help create jobs in the private sector since the Workers have?

Except for the CCC projects which worked on mostly state, local and federal government turf, all of the other...the big projects... were all done by private contractors. While none of these projects were truly big enough given the size of the problem, they WERE big enough to prevent the depression from falling to a point where any recovery was problematical. In the bigger picture there were some strong movements such as the Nazis and the communists that if there had been no federal response to the depression one or more of these groups could have, and probably would have attempted to stage a coup 'd etat. We can't even imagine the result of that kind of outcome, though Sinclair Lewis wrote a book called 'It Can't Happen Here' where in fictional form...it did happen! FDR managed to keep the greater part of the population believing that the New Deal was far preferable to a total overthrow of the federal government and the substitution of either a communist regime or some form of European fascism. Historically, it was a near run thing. Russia was very active in promoting an American turn to communism and Hitler and Mussolini with their American allies in the German American Bund and the KKK believed they could organize a far right revolution...and weren't afraid to say so! The war knocked those ideas off the national agenda and the post war boom that included the creation of a labor union led blue collar middle class teamed up to save the day. That's why today there aren't any commies or socialist with any real influence in national politics, though the current GOP/Tea/Fox/Jesus party defends a form of corporatism that isn't terribly far removed from the Italian fascism of the pre war and war years of WW2.

Government projects use private companies for basically everything except spending and collecting the money. Stop and really think where that concrete, paint, trucks and skill workers are coming from. The reason the government is bloated is becaus they overpay and they requisition cash that turns into a slush fund.