Recently naturalized U.S. citizen here... airlines or aviation companies currently do not sponsor visas for residencies to pilots. There are way too many pilot applicants for aviation jobs.

In order to sponsor a visa, the sponsoring company has to "prove to USCIS" that there is a shortage of applicants in the field. Flight schools sponsor visas but they are for flight training, not for US residencies or employment with airlines (read as 'green card holders' or 'U.S. citizens' only). You might find an employer who is having a hard time finding ground school instructors, or for some other aviation related job.. and they might sponsor an employment visa, but then you wouldn't be flying commercially in the US.

An employer would have to sponsor you twice (3+3 years), and after you've lived in the U.S. for 5 years, you would be eligible for a green card. If you got married to a U.S. citizen, or a green-card holder, they could sponsor you... but you have a girlfriend, so that's out. There's the green card lottery, if your country is eligible but I wouldn't hold out on that. Unfortunately, right now, your options are limited. If there's a massive shortage of pilots in the U.S. over the next few years, things might change.