“The time that I’ve had away from the game has been a really good thing for me,” the 31-year-old said.
Resuming on 440-8, Nathan Lyon added just one run before offering an easy catch to substitute fielder Dinesh Chandimal at mid-wicket off Angelo Mathews in the third over of the morning.
Bird failed to get off the mark in his first Test innings and was bowled by Shaminda Eranga, leaving Johnson stranded.
Clarke scored 103 for his 22nd Test century and first at the Melbourne Cricket Ground, and earlier became Australia’s highest-ever run scorer in a calendar year.
The Australia captain has 1,595 runs for this year, eclipsing recently retired Ricky Ponting’s previous record of 1,544 in 2005.
Watson, who has a career 18 Test half-centuries, fell short of his third century when he fell for 83.
Glenn Maxwell will replace Watson for the Sydney Test after the all-rounder again strained a calf muscle while bowling on Monday.
Usman Khawaja has been placed on standby for Clarke, who remains troubled by a chronic hamstring injury that had made him doubtful