The A320 first crashed on literally its first passenger flight "(in service for two days)". https://en.wikipedia.org/wiki/Air_France_Flight_296
Later, an A320 crashed less than two months after delivery. https://en.wikipedia.org/wiki/Indian_Airlines_Flight_605
These two crashes, as well as a third flight later, Wikipedia describes as "a series of crashes caused, at least in part, by what was believed to be pilots' unfamiliarity with the sophisticated computer system of the Airbus A320". https://en.wikipedia.org/wiki/Air_Inter_Flight_148
Different sources disagree on the causes of these flights. Most indicate "pilot error", but the pilots themselves and other organizations dispute these reports and instead claim design flaws with the aircraft. In general, most discussions of the accidents focus on the new fly-by-wire system.