Applying to US PhD during first year of Master's program I am in the last year of three-year Bachelor's program in Computer Science in Europe. I am interested in doing PhD in the US, but I did not apply for it this fall because I did not have any sensible research, nor strong recommendation letters by the deadlines (though I have perfect GPA).

This summer I am going to do a research internship, and for the next year, I think to enroll in Master's program (two-year) at my current university and work on a project in a research institute (being enrolled is often required to do any project or internship). But I am not interested in finishing that particular Master's program.

I believe I may have a relatively strong background by the next fall and would try to apply to grad schools in the US (and probably non-terminal Master's programs as well). However, I sometimes hear that admission committees frown upon applicants who are already enrolled in other programs and some do not admit students with unfinished degrees at all.

Can having unfinished program really hurt my application? Would it be better if I delay my graduation from Bachelor's program instead?
I can't speak for computer science (my experience is in Math), but I wouldn't regard it as a serious issue.  Typically programs are more concerned with your general level of preparation (for example, the content of the courses you have taken) rather than the formalities of earning degrees, especially in a foreign system which can be a bit confusing anyway. Certainly if I saw your application and it looked good otherwise, I wouldn't have a problem with you not finishing your masters program.