This is government spending on a private sector endeavour. It isn't like government went out and hired individuals to perform the bridge building. The companies hired are all private sector. The one contractor, with whom the government chose to deal, is the one that explicitly stated that they would be out-sourcing a large percentage of the job.
This is NOT a private sector project. It's government all the way.
The government must extract wealth from the people that generate it (profit) in order to do what it would like. Government usually pays corporations to do what it wants but that simply builds a bigger cake with the same base. All you do is move money around to different pockets, but no new wealth is created.
Something of this size can seriously alter markets as well. LOTS of steel and concrete being used here, which increases the demand for these items and lowers the supply. This affects other parts of the economy in what is known as opportunity cost. Because government has access to so much capital (spending on things that the market doesn't want), it is very easy for it to seriously skew the market.