To think shut the hell up? (New manager)

Tell him!

Tell them. It sounds like it’s pretty clear.

Time to go back to your old manager, his manager, tell them the situation so that they can put him / her straight!

Clearly remind him that this is THEIR workload, that you were only helping by covering some managerial tasks while someone was being recruited and that you are sticking to you own workload.

I would put that in an email and copy in their manager.

They are new to the role and on probation so you want to nip this behaviour in the bud and make management aware of what is happening.

Remind him that it's the work he was unable to cope with as a new starter and that you have been helping him out, not the other way round. You need to set your stall out early and make clear that you won't accept his bullshit. Also, get the rest of the additional workload off your hands and into his ASAP.

Is he under the impression that he doesn't have to do any of the actual work?

I wouldn't word it quite like that.
She didn't take it off his hands because he couldn't cope, more that he was getting up to speed in his new role.
Was OP really helping him out, or was it an expectation of her role?

Clearly the time has come for new starter to take that workload. So:

"No, I was not helping you out, my role was to do that work until you were settled in. It is and always was intended to be part of your workload and now it's time for you to take responsibility".

If he pushes back then maybe suggest you both talk to the more senior manager together.

@justanothernamee If you're on pretty good terms with your old boss, just have a brief chat and explain that this prick thinks he doing you a favour by doing some of his own job. Ask them to explain to him in little words what his job is.
Good luck.
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Lots of bad advice here! The manager above your new manager is not going to have allegiance against their immediate subordinate on your behalf. This is one of those times you have to suck up comments and move on. They like to hear themselves talk and play fantasy and it's your job to just nod and pretend you agree so that they think you're on side. Too many people think that "speaking up" does anything in these circumstances but all it will do is single you out as someone who isn't a "team player" and who is easily agitated by words. If your job was made easier, who cares what they say about it? The actions speak louder than their words - or should do.

I don't understand this.

Catch ups work two ways so you've obviously told him that it's his work and not yours?

What does he say when you point it out?

If he still thinks it's yours after the catch ups, you'll have to get his manager to explain it isn't.

There's no allegiances - the new starter needs to be reminded either by OP or (if that has no impact) by their manager.
It sounds like you don't work in a very good environment if you believe speaking up when things are not right is regarded as not being a team player or easily agitated.

I think I work in a realistic environment so I know how corporate management works. Ideally, we'd speak up and things would get sorted but there are office politics and this is not the way to address them in a way that will get the OP the result they want. It's a comment. The action/result that the OP wanted has been done so why make a big deal about what someone is now saying that has no bearing on anything apart from feelings? Corporate doesn't care about feelings but they care greatly when you try to make a big deal out of things they think have been sorted. Experience from working in corporate for over 20 years and making my way up the ladder. It wasn't accomplished by being a pushover nor not being strategic about what to react to vs what to tolerate for my greater good.

Not everyone's a wet lettuce.

Plenty of employees have the gumption and professionalism to speak up when necessary.

However, the OP hasn't mentioned her side of the conversation during the one to ones, so it would be interesting if she came back and told us about them.

@LintRollers I'd say anyone that is reactive to words is a wet lettuce. Hopefully, OP will learn that actions speak louder than words.

Speaking up for oneself when something is clearly wrong is always the right thing to do.

Although sadly many employees (particularly women) don't have the courage to do it.

And calling an employee a wet lettuce for being reactive to words, would be seen in most places of employment as downright bullying thankfully.

I am absolutely not doubting you @justanothernamee but why didn’t your employer open up another role same as yours if the need was for more hands to do the work? The new bloke might (reasonably) expect or have been told that he would manage you and a lot of managers aren’t expected to be hands on except in an emergency for example. I’d seek clarification from old boss - what does his role look like when he’s learned the business? If he is supposed to say do 50% management and 50% ‘same work you do’ then no, he is not supporting you. Supporting you would be when the workload goes up
and he has to do more of ‘that work’ during a busy spell, say 75% of the time doing what you do.

If they have told him something different, someone is going to be disappointed, right? Seek clarification first (stay calm, get the facts first).

I don't know exactly what you mean by corporate management.
OP states she works in a business with a manager and they have catch up meetings. I am not sure how you have managed to come up with the specific, unhealthy work place you're writing about.

Speaking up and saying "Errr, that's not my job" is hardly making a big deal out of anything.

What's 'reactive to words'? Hearing something and having a response? In this case OP is cross (is that what you mean by reactive?) that her manager is implying he's helping her out.

You need to professionally correct the issue. Explain that you temporarily picked up those tasks while they were onboarded with the organisation, but they in fact fit in his role. Before you do that - how are these comments being made (your 1 to 1 or just generally) because I would be inclined to document it and then seek clarification from the previous line manager.

You want to handle it delicately.

I'd play it slightly differently than just calling him out on it. Ultimately he's you're new boss, he's going to be doing your performance review etc. I'd ignore the comments when he makes them, but at your next catch up, do a "I think I've handed back all of the bits of your role now, other than XYZ (whatever is left), I just wanted to check you're picking up the full role description now, I don't want anything to get left in the cracks, i would hate for something to get missed if i don't hand it back to you properly?" It gently puts him in his place. To be fair to him, you don't know what his boss/your old boss said, she could have said "it's earlier than planned but OP isn't coping, ad a favour can you pick it up ahead of the timeline please"