For example, the English Premier League this year finished on 23rd May and starts again on 13th August (82 days rest). Last year, teams played 9-10 games between the COVID resumption on 19th June and league finish on 26th July. There were then just 48 days rest until the next season started on 12th September 2020.
The continental competitions (Champions League and Europa League) were also played during August 2020 meaning many players at top clubs had next to zero break between the two seasons. This has all had a knock-on effect to this tournament. I'm actually surprised there haven't been more injuries.
P.s. Small nitpick but the "European Cup" is the aforementioned Champions League trophy which is contested by the top club teams in Europe. The current international tournament is the European Championships or Euros.