The original Death Note never cared about romance. Instead, it just comes off as a clumsy plot and a scatterbrained, careless main character. You might assume Turner wants to get caught, but if he does, then none of his other actions make any sense. He hides the damn thing in his calculus textbook, never denies being a Kira supporter, and outright tells L where it's hidden during one of their only confrontations. One of the first things Turner does is show Mia Sutton the Death Note, with no foreknowledge whatsoever about how she might react.
While both characters need not be entirely identical, it's hard to care about what happens to Turner when he obviously doesn't. Eventually, not being caught becomes his primary goal, while actually pursuing his interpretation of justice becomes secondary. Later, he lets himself be handcuffed to L for months on end to allay his suspicions. In early episodes, Yagami hides the Death Note in a false bottom into his desk drawer, which he rigs to explode if tampered with. One of the primary conflicts in the original Death Note is Light Yagami's increasingly frantic struggle to avoid being caught.