I’m wondering if sites like the UK database one (I won’t put the actual url, as linking out gives sites more publicity and more power), could actually be stopped with GDPR. Where it may violate GDPR, is the fact that it calls itself a database of criminal reports for a particular subset of crimes. It also encourages people to browse and search by area, and lists results by the name of the offender. It is one thing for a newspaper to have archived articles online generally, but when someone repurposes (most likely plagiarises) the content to make a “database” of a type of criminal convictions, prioritising listing names and locations that is another matter.
GDPR states:

“Processing of personal data relating to criminal convictions and offences or related security measures based on Article 6(1) shall be carried out only under the control of official authority or when the processing is authorised by Union or Member State law providing for appropriate safeguards for the rights and freedoms of data subjects. Any comprehensive register of criminal convictions shall be kept only under the control of official authority.”

They are quite clearly not working under an official capacity, and although much of the content is or at least was in the public domain, as they just repurpose content. I’m not sure that is a defence. They are certainly trying to put together a comprehensive register, and even encourage people to report cases.
I think there could be scope under GDPR to challenge this, but it takes time and money, and most likely will just give them more publicity. So I’d query if it is worth pursuing? Another angle could be copyright infringement. They seem to just copy word for word newspaper articles. But it would have to be the copyright owner that requested action, and I’m not sure newspapers really care. So many people are negatively affected by these sites, not only ex-offenders but their families including children. I do wonder if some coordinated action could take place. Interested in thoughts about this.