When the detectives are interviewing someone or working a crime scene, they are never seen taking notes. Real detectives are constantly taking notes. The notes are so important that they are occasionally booked into evidence to ensure the originals will be available for review before trial.
Multiple cases are shown which have interstate or international components. While NYPD SVU detectives might cooperate in those cases, they would primarily be handled by the FBI or some other federal law enforcement agency.
The time-lines of trials seem to be rushed and take place within days of the crime. Most of the cases on this series are rapes and homicides, with a kidnappings thrown in. These cases, if not plea bargained, are seldom heard in less than a year after the event.
Despite many of the crimes being homicides in addition to being sex crimes, homicide detectives are not shown to have responded to the case. The major crime would be the murder itself, with the sex crime elements being included offenses.