There are a couple of reasons:
He wasn't thinking
His fist action after Sirius's death was to try to kill Bellatrix; he wasn't logically thinking at that moment. He wouldn't have thought about using Time-Turners.
Sirius would resist.
If Harry appeared randomly and tried to save Sirius, he would be suspicious of some kind of Dark Magic. Or, if he didn't get suspicious, he would turn the tables and try to protect Harry, at which point nothing would be different.
He didn't have a Time-Turner
Hermione gave hers back at the end of 3. Yes, there were Time Turners in the Department of Mysteries, but:
...while the glass cabinet that Harry now suspected had contained Time-Turners continued to fall...
“Ar, I always knew yeh’d find it hard ter squeeze me inter yer timetables,” he said gruffly, pouring them more tea. “Even if yeh applied fer Time-Turners —”
“We couldn’t have done,” said Hermione. “We smashed the entire stock of Ministry Time-Turners when we were there last summer. It was in the Daily Prophet.”
Even if he had thought of it, he couldn't have. Even if he could have, it's unlikely that it would have worked.