The EU was not set up to be a single organisation like the USA. It was set up so that a group of countries, could remain independent, and yet still work together.
Some people would like it to pull closer together; Some people would like to pull it apart. However, the fact remains, that it is not, and it never has been, a single country. It is a group of countries working together, and it's institutions reflect that.
The Primary Institutions of the EU are;
The EU Council is the Primary House of the EU. That means it's like the UK House of Commons. However, because the EU is a group of countries, and not a single country, it works slightly differently to the House of Commons.
For a new law to pass in the UK House of Commons a majority of MP's must support it. For a new law to pass in the EU Council, every single member has to agree. That means that no group of countries, can ever force EU laws onto any other country.
The EU Council is comprised of the heads of state of each of the members of the EU. So the current member for the UK is Boris Johnson. If we have new elections, and we elect a new Prime Minister, then they automatically become the UK member of the EU Council. The same is true for all the other members.
As such, each country elects their member, in the same way that each constituency elects their representative (MP) in the House of Commons. The EU Council is therefore directly elected, and very democratic.
The EU Council is basically the most important chamber, and NOTHING happens as far as new EU rules or laws goes without everyone in the EU Council agreeing to it. Every country can just say "nope", right up to the point where a new law is passed, if they don't like it.
However, heads of state are pretty busy people, so they don't tend to do all the work themselves. As such, every head of state can appoint an EU Commissioner to represent them and their country.
The EU Commission is where all of the busy work gets done. It's where legislation is drafted. It's very much like the UK Civil Service. It's not directly elected, but EU Commissioners are directly appointed by heads of state.
No one ever complains that the UK Civil Service isn't directly elected, and why would they. It's down to the government to appoint them. Same goes for the EU Commission, and every country has an EU Commissioner.
The EU Parliament is directly elected and has been since 1979. It's where MEP's sit. It is there to conduct an additional level of democratic oversight to the whole process.
This next bit is important, so if you've been skim reading to this point, slow down, and make sure you take this bit in.
The EU Parliament cannot propose new legislation. It can only vote on whether or not to allow legislation which has been proposed by the EU Council, and EU Commission, and here's why;
If the directly elected EU Parliament could propose legislation, then the EU would be acting like a single country, and not like a group of countries.
It would make it possible for the EU Parliament to become more powerful than the EU Council, where the heads of the individual countries sit.
As such, only the EU Council, or the EU Commission, at the request of the EU Councilors, can propose new legislation.
This is to stop the EU acting like a single federal United States of Europe, and to ensure that it always acts like a groups of countries working together. Power resides with the EU Council, and not the EU Parliament, to ensure this is the case.
So the next time you find yourself arguing that the EU is not democratic, it's primary institutions are just as democratic as the UK's, if not more so, given both it's primary and secondary chambers are directly elected. In the UK only the House of Commons is elected; There are good reasons why the House of Lords is not elected, but that's a totally different story.
What's more, in EU elections, proportional representation is used, which means that far more people actually get a say in choosing their representative. In the UK elections of 2015, UKIP received nearly four million votes, which under proportional representation would have given them more than sixty MP's, but under the UK First Past The Post electoral system UKIP got zero MP's. Whatever your opinion of UKIP (And I really don't think that much of them myself), I think you have to accept that is something that UKIP voters have got very good reason to be upset about.
So unless you actually want the EU to act like a single country, maybe appreciate why it is set up the way it is; To be a group of countries working together, where non can force the others to act in ways they do not wish to, and where it's Parliament doesn't have the power to change that.
Unless you want it to become a single country of course.