Parkrun organise free, weekly, 5km timed runs around the world. They are open to everyone, free, and are safe and easy to take part in.
Park Run happens every Saturday at 9:00am at 3 locations in Norwich:
Eaton Park http://www.parkrun.org.uk/norwich/
Colney Lane http://www.parkrun.org.uk/colneylane/
Totally free to take part, but please register before your first run.
They have park runs all over the UK, not just Norwich.