How can I transfer money from a US bank account to an IBAN+BIC number (European bank account) without any fees?

Franck, "a rate" does not exist. Dealers who exchange 10s of millions, complain bitterly that other guys who exchange hundreds of millions get a better price. Rates are incredibly different up and down the scale of "how much you are changing" and it also varies quite dramatically (per that scale, ie the curve changes) depending on particularly volatility, time of day etc.

Beware of banks that (incredibly stupidly) claim they have "no fees!" or "no commission on 'the rate'"! Seeing point 1, there is no "rate". There's no official, standard, or whatever rate. Banks that offer "no points on the rate!" (idioticially) sit down and write "the rate" and then assert they are offering "just that rate!!" It's completely stupid.

So note to begin with you are trying to sell some USD and get some EUR. That's much like say trying to sell a house. There will be a multitude of offers.

In general terms some banks generally and usually give you better rates than others. For example it's no surprise and widely known that HSBC is generally "surprisingly good cross rates". (Since they are the world's major and primary money laundering device for politicians and also minor villains such as terrorists and criminals, maybe that's the reason, IDK.) Another example (you mention Lithuania) in broad general terms Swedbank (or whatever they're called now, who knows) is "known to" give, generally, good cross rates.

Thus it's really just a matter of (A) essentially investigating (which takes forever) which bank is offering a good rate on your particular Quad of currencies and countries, and indeed (B) just kind of "knowing the gossip" on which bank is best for that Quad.

Then a big issue is OFTEN (but not always) if you are intrabank, it is much much better. If I would change, you know, currency X in my HSBC act in country A to currency Y in my HSBC act in country Z - that is often (but not always) better.

Multicurrency accounts! Here is a tip. It's usually better to change currencies within your one account (ie in your bank in one city). As you know, multicurrency accounts are unknown in the US but in Lithuania etc everyone has one. In the example I would send USD to Lithuania and only then (perhaps later) worry about converting it.

Companies that exist to do precisely what you are saying. I have always gotten great rates (/fees overall .. whatever) with ofx and transferwise (I prefer ofx but whatever). They are the only 2 companies in that category. However, lately a number of people have told me "oh their rate sucked when I checked" or "the fees suck". So IDK. Either those People got confused and compared the rate wrong, or times have changed - or whatever. But in the first instance you'd wanna check that - that's your basic goto. Give them a ring.

Companies that exist precisely to exchange money! In places like Lithuania, UK etc there are very much companies that just exchange money (I mean not cash on the street, but large amounts of money by wire etc). As distinct from item 8: item 8 are companies which both "transfer and exchange", ie, they have literally built a (worldwide) infrastructure "for moving money". In contrast item 9 companies literally just "trade in currencies." You would have to carefully find item 9 companies (in Lithuania) and ensure they are a big major player. In this case you just wire them the money (they don't help you with that in any way, you have the usual hassles/costs), they swap it and send it to your local (ie Lithuania) bank. These guys will get you the BEST rate, but no help at all on xfer fees. (In contrast the "8" people, which are worldwide operations which handle both ends, the "9" people are literally just companies in Lithuania (or the UK or whatever).) (In the UK in particular there are many such "currency specialists", particularly due to the many folks in the UK buying properties in Spain, Croatia etc etc.) (Well, they used to.) So you need that sort of thing in Lithuania.