Spring (April to June) and autumn (September to October) are the best times to visit — the weather is good, the light is warm, and the crowds are manageable. July and August bring summer tourists from the Costa Brava and Barcelona, but the medieval streets are dense enough to absorb them; it's busy rather than overwhelming.
Weekday mornings are the least crowded at any time of year. The tour buses typically arrive mid-morning; if you get there early, you'll have the bridge largely to yourself.
The Mercat Medieval (Medieval Market), held over a long weekend in mid-September, is Besalú's major annual event. The town dresses in period costumes, the streets fill with market stalls selling crafts and food, and there is music and re-enactment throughout. It draws large crowds but is genuinely good — the setting is built for it.