Police in Camden, N.J., are fielding more calls for overdoses than for homicides this year.

Fatal opioid overdoses this year in the city of about 76,000 had surged to 35 as of Friday afternoon compared with 10 at the same point in 2016, and will likely eclipse last year’s 12-month total of 40 deaths within weeks, according to Police Chief J. Scott Thomson.

“The...