If you are applying for positions where you need recommendation letters, it's going to look pretty strange if you don't have one from one of your co-supervisors if it's possible to tell who they were. You don't say what degree you are currently working on, but I'm guessing it's either your undergraduate or master's degree since you appear to be applying for a PhD. I wouldn't recommend saying the following directly to this person, but refusing to write a letter of recommendation in order to try to prevent a student is highly unethical.
I recommend you talk to your other co-supervisor and see if you can get them to help you talk the recalcitrant co-supervisor into writing you a letter. It is very common for students to move from group to group and institution to institution while getting their successive degrees. In fact, most institutions in the US have a preference towards moving around like this. While losing a good student is hard, it's very inappropriate for a professor to try to prevent a student from moving on.
If the professor is having trouble coming up with things to say about your research progress, you can help him out by writing some text about what you've done so far and the good progress that you've made. Faced with some true statements about how good your research prospects are and the progress you've made to date, they may cave easily and write you the letter you need. If you show your resolve to apply elsewhere, they should hopefully give in.