Light's Search History Death Note
Primarily opinionated question the way its phrased. However Light did rational this in several moments when he planned to hide his notebook. 

He followed a line of thought that even if someone found the notebook, he could easily say he was researching the deaths for his planning to follow his father in law enforcement. Same can be said for his Internet or browser history. 

Even if you can pinpoint Light searching for criminals before their deaths, it doesn't conclusively prove that he is indeed Kira and not some over enthusiastic student that just wants to research criminals. Remember Light doesn't immediately kills criminals but spreads them out randomly and kills on average similar number of people. While he may be researching more of them.

The show is all about the main detectives L, M and N somehow intuitively guessing that Light is Kira but conclusively proving he is indeed Kira is what makes the story interesting!

Finally, as I've said before multiple times. When you watch a fictional story you've to accept that the author might've tried his best to stop plot holes, but at the end of the day they will exist. The death note and shinigami don't exist. Overthinking things and creating plotholes will reduce the enjoyment of the genre. You've to assume some things that the author does and listen to the story he/she wants to tell

Cheers~