Regardless of the GDPR, every European country has an independent public authority responsible for providing advice on a legal term regarding data protection issues. Those authorities are also responsible for prosecuting any violation of GDPR requirements, under their own national territory.
Essentially a Data Protection Authority (DPA) supervises and investigates cases of data privacy violations and enforce GDPR fines.
Just between 2020-2021, we have witnessed many cases of GDPR enforcements by DPAs, such as Google in France (€50 million fine), Jö-Bonus club in Austria (€2 million fine), and Vodafone in Spain (€8.15 million fine).
You can find the complete list of European authorities here with their contact information and statistics of their imposed fines.