There is a language/cultural barrier between me and my supervisor, what can I do to improve this situation?
It will be much easier for you to learn German to conversational level than all of your co-workers learn English. Even if they are some-what able to talk in English, conversational English is actually pretty tough, whilst German is pretty structured and easy once you know the rules (and the many exceptions ;) )

More importantly, you moved to Germany. Whilst English is expected of all Scientists, there is probably more of an expectation in your lab that you will, eventually, learn German.

Personally, I would tell everyone you are going to German classes on Monday mornings, etc, and that should not only fix this issue, but also give you a positive perception among your co-workers as someone who is a fixer.

Also, speaking from experience as someone who didn't take their own advice when doing a PhD in Germany - if you dumb-down your English for non-native ears, you very quickly find yourself not being able to speak English with natives. Whenever i go back home my parents think I have some kind of brain damage...