Steve Bruce admitted his job is in jeopardy after Sam Byram's lucky leveller gave Norwich an undeserved point.
Dara O'Shea's early header had put Albion on course for only their second Championship win of the season.
But fortune deserted Bruce and his players in the 68th minute when Teemu Pukki's shot went in off the unwitting Byram — and Albion appeals for handball failed to convince the officials.
Results elsewhere left Albion firmly part of the relegation equation going into an international break that often sees under-performing sides part with their managers.
Bruce was adamant referee Thomas Bramall and his officials had erred with Norwich's goal however, as well as getting an earlier penalty claim wrong.
'I know it's a hard job but big decisions have cost us — and if I don't win games, I get my head chopped off,' Bruce said.
'The big decisions they need to get right and we had two against us — their goal and the penalty — and that is being genuinely honest.'
Albion caught Norwich napping in the eighth minute after Marcelino Nunez fouled John Swift on the left of the Norwich box.
Swift took the free-kick himself — and the home defence simply stood and watched as O'Shea cantered in unchallenged at the back post to plant a downward header beyond keeper Tim Krul from five yards.
Fortune favoured Norwich a few minutes later, when Kenny McLean's wayward pass allowed Albion to send Swift into the box, but referee Bramall failed to point to the spot when Grant Hanley appeared to barge him over.
'It was an absolute stonewall penalty, it doesn't matter that the ball was going out of play,' Bruce insisted. 'Swifty got a touch, the boy Hanley barged into him — it's a penalty.'
Dean Smith, the Norwich manager, argued that Bramall had to let play continue as he had allowed a similar challenge on a Norwich player to go unpunished in the build-up.
'It's consistency from the referee, so well done,' he said.
It took the home side an age to make any reply but Nunez hit a first-time effort from a half-cleared corner inches wide to rouse the Carrow Road faithful.
The Chile midfielder had another go in similar circumstances a few minutes later but comfortably cleared the crossbar this time.
Norwich fans had been less than enamoured with Albion's reluctance to take free-kicks or throw-ins quickly, and the disapproval with Bramall for not enforcing a speeding up process grew stronger as the second half progressed.
Josh Sargent had been kept quiet at the other end, as had Pukki, but the pair combined for a shot from the former that was deflected behind off a defender.
Fortune deserted Albion from the corner as Byram's fortunate finish — his first for Norwich — was allowed to stand despite claims that an arm had been used by the defender before the ball crossed the line.
Onel Hernandez laid the ball back to Pukki for a shot from the edge of the box that wrong-footed David Button by taking a massive deflection off his team-mate and, although referee Bramall talked at length with assistant referee Matthew Wilkes about Albion's handball claims, it was still allowed to stand.
With no VAR to help or hinder, it remained a huge debating point but Smith again backed the officials. 'I'm led to believe that if it's T-shirt — if it hits the top of the arm — then it's OK so it looks like a good goal.'
The result ended a six-game winning streak for the Canaries in league action.
Though the draw leaves Bruce's Baggies in an uncertain position — for both team and manager.