Internships (in the US at least) are generally targeted towards people who have no industry experience as a way to get experience that is low risk to the employer. Given that you have four years of work experience that is going to put you at the bottom of the pile for prospective internships.
Keep applying for jobs while working on your portfolio. Even if it's not games, that four years of experience is better then a fresh graduate with just maybe an internship.
I looked at a lot of linkedIn profiles for programmers in positions I want to reach and many of them started with an internship then continued working as juniors for the same company.
So I'm not sure that my experience is better than a graduate with an internship.
In the meantime I get a lot of rejections from companies saying I don't have the right industry experience.