Yes, defense counsel would have access to the evidence the prosecution is using against the defendant, and has powers to get other relevant evidence. And they would make that argument. In a jury trial, web search history would usually be presented by a police investigator. The prosecutor will ask him questions to elicit the picture the prosecution wants to paint, while defense counsel has the opportunity to cross-examine the witness to poke holes in that picture.

Say the investigator testifies that the defendant searched for: "how long does it take for a dog to die in a car?" On cross, defense counsel would definitely point out: "but the defendant didn't search for how long it would take for a baby to die in a car, and he did have a dog he might have been worried about leaving in the car."