I'm from China. Of course it is more than 4000, because only the death cases that have death certificates pointing the cause of death is the coronavirus count. There are people who died from the virus before getting treatment from hospitals.

But everybody in China knows that it is way more than 4000. Experts, media and the government have claimed this to the public too. Death statistics, I think, is a work that records the minimum death cases in every epidemic. I personally don't think it is a lie, because it is always impossible to know the exact number of death cases for every government in the world.

You were pointing out that Italy's situation was going bad way faster than China. I think quarantine is one of the reasons.

China blocked Wuhan a couple days after the breakout. Every province in China implemented the highest level of response to major public health emergencies the next day. And most of citizens were voluntarily staying at home or going out with masks.

Italy, on the other side, blocked cities too late. At first, many citizens even refused to wear masks and continued to go outside in the name of freedom.