I can go back earlier mate I just picked the first, examples - 1st October;
2nd October;
19th October
I mean, the fella had some pedigree. He had us in Europe. We backed him but didn't get him a striker. This was the start of his Prem;
Man City A (L) Chelsea A (L), Tottenham H (L) Man Utd A (L), Bournemouth H (W), Burnley (A) L
Koeman had a lot of mitigating factors which I shared at the time here:
https://www.grandoldteam.com/2017/09/11/early-days/
But fans, including yourself was screaming for his sacking very early into a new season so you can't go laughing off the thought of fans wanting a manager out after 5 games as silly.
The same will happen again next season if he lost the first 5.
Managers don't get that luxury of time. [Edit - if results are that bad/successive defeats]