The whole representative system needs to be torn down and built again from scratch.
Begin by creating single lists for the European Parliament for the whole EU, instead of the current system, where every country votes for their national lists from their national parties, which then get together with affine parties from other countries and form a parliamentary group. Create common European parties/lists and common European candidates.
Make the EU Commission, or however we'd like to call the reformed executive branch, directly elected. This of course problematic, since dual elections are typical of presidentialist or semi-presidentialist systems and the most common system in Europe is the parliamentary one. If my first suggestion is fully undertaken, we can let this one go.
The European citizens cannot feel attached to nor feel they are accountable for the decisions made in Europe and Strasbourg if they don't feel like they have elected the people who are supposed to represent them.
In short, make the EU a European Union with a European agenda, not a gathering of national sides with national agendas playing give and take.
The EU's biggest flaw is the lack of accountability. The unelected European commission has more power than the elected European Parliament. The easiest solution would be to transfer power from the commission to the parliament so voters can be held responsible for their choices.
Regarding common European parties, they do exist but nobody votes for them. People will always prioritize their family and local surroundings over the common good. There's a reason even national elections consist of many local parties.
In any case, the point of the EU should be free trade and dialogue between neigbouring states, not the creation of a Federal Europe.