New York City’s libraries and other cultural organizations looking to build facilities or make repairs often endure a staggeringly long process and added costs under the agency that manages such city-funded projects, a report finds.

The city’s Department of Design and Construction was created in 1996 to consolidate and more efficiently manage the city’s capital projects. But a study issued by the Center for an Urban Future, a nonpartisan policy think tank, found that delays and cost overruns were pervasive among the 144 library...