Actual construction is performed by contractors, corporations that are contracted by the government. In the US, it is rare for a construction project to be built, or even be designed, by government workers. These corporations and their workers can be sued, and take on additional liabilities due to the contracts and laws that apply specifically to them.
Which is to say, you are spot on. Most of the work government engineers do is oversight. Oversight is really important.
I've constantly read that in the US due to ideological reasons the agencies that handle oversight are chronically understaffed which results in delays and inefficiencies. Delays are a big problem in the US. The slower a project is done the more it costs. Without proper oversight, you have an inability to efficiently manage design changes leading to delays and or less efficient construction.
What? Two women suing McDonalds because they got fat from eating too many cheeseburgers is not a regulatory issue. A canoe company being sued for not having lifeguards posted every mile of the river they serve is not a regulatory issue. Parents suing a city because their idiot kid fell off a slide is not a regulatory issue.
America is lawsuit-crazy. Anybody can sue anybody for anything, regardless of whether regulation surrounds the issue or not. If anything, regulation acts as a legal protection against lawsuits, and many regulations are enacted in response to frivolous lawsuits from assholes who think the world owes them something because they can't take personal responsibility.
I always say there are not too many lawsuits, but too many laws. Legislatures think they are making something better by passing more rules, regulations, and laws. But, for every rule, regulation, or law, you have an opportunity for a lawsuit. Use more common sense? Sure, but then don't pass a multitude of laws regulating the subject.
Why are infrastructure projects' costs higher here? I don't know really. My guess would be that compared to say Germany and Japan, everything here is the U.S. is huge. I mean probably 10-30% larger in size than those other countries with the resulting higher costs. But, this is just my personal observation.