they are just f.cking lying and i hate it
It's the prisoner dillema. They're lying about Sangwoo fessing up to make Bum tell something he knows. Either Bum might think Sangwoo told them about him killing the woman back then, but I'm not sure if Bum will betray Sangwoo even if he does think that way.
That's what I thought at first too, but Sangwoo may also have just "confessed" by saying that Bum killed her, since he technically did. He may be trying to pin everything on Bum.
The police wanted to trick them and I think that's exactly what's happening. They have no evidence whatsoever yet and Sangwoo knows that. He wouldn't feel as cornered to come up with such a desperate alternative...atleast not yet. He would be going strong with his earlier story about them being a couple and him acting out to protect Bum.
Did he actually confess?!! What? Why?! Did he make something up?!! Don't just end it there! You're torturing me T_T