edit: I also found out that with a LM741, that problem does not occur.
edit: First of all, this is not a "should I use LM324 as a comparator or not" question. Besides that, I just wonder the reason for this behaviour. I will paste one of my comments also here, since I think it describes the part I don't understand better.
However, I don't understand why it is related to opAmp's capability of resposing fast. The output goes 0 as soon as V+ goes below V-. So it looks like it does not have a problem with catching up. Rather, it, not understandably(to me, at least), goes high again while no change occurs in the state of V- to V+.
Why does not output stay at 0 but rises again high? I am talking about the yellow rectangle portion. The signal at the non-inverting input is higher than Vref on the inverting input for that duration. I would expect it to stay at GND as long as V- > V+.
I am using LM324 model from Texas Instruments Website.
Update : For all who might be having difficulties in such applications, I figured out this will be a better solution, after the answer of Marcus Müller.
Answer
You're getting phase reversal because you are taking the input below the negative supply rail. Add a negative voltage rail (-9V) to the op amp and it'll be OK.
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