Nope. Not normal. She is totally treating that relationship like she's still the wife. If he's allowing it that's not good. He needs to knock that shit off.

And what does your partner do about it? Does he go to the cinema with them? Do the shopping for her? "Because it's for the kids" ... or ... "to keep the peace?"

If he's doing those things, then it's more important to him that his EX is happy than YOU are happy. Won't tolerate conflict with the ex, but will tolerate conflict with you. That tells you all you need to know.

That is not ok. They are still acting like husband and wife. I'm so sorry.

If she is still asking after you have been together 4 years, i'm guessing he has been doing these things. 4 years of saying "no" should cause her to get the message. Your partner has a responsibility to shut that down if he values your relationship. Now, i am of the thought that this is not something you should have to spell out for him, like "doing these things with/for your ex is putting her needs before our relationship because you are giving attention and help to her that should be reserved for the woman you are currently with. You can have a good relationship with your kids that is separate from the relationship you have with their mother. It bothers me. Please stop immediately." Sadly, from experience and from reading this board, there are some selfish or just plain dense people out there that will try to make you accept all kinds of degrading situations in the name of their prior relationship kids. But you won't know which one your partner is until you spell it out for him. 

Your BF needs to put a stop to it. It's not appropriate for her to ask but it is especially not appropriate for him to do those things.

Nice to welcome someone from Devon where I grew up!   In answer to your question - damn right it is not appropriate!  If I discovered my DH had been going to the cinema with his exW his bags and chattels would be outside the door, locks changed, quicker than you could say Jack Robinson.  

I remember a few months after DH and I got together, his exW asked him if he wanted to accompany her and the SDs on holiday to France.  I told him, if you are even considering this, we have a big problem.  Needless to say, he didn't go. 

I'd have an issue with the movies. Picking up something from the store not so much.

Everyone has different relationships post divorce. It's not right or wrong.  If you're uncomfortable with his it's best to move on. 

This line above rang with me, "He won't tolerate conflict with the ex, but will tolerate conflict with you. That tells you all you need to know."  If the idiot can't figure out who wife #1 or GF #1 is after four years, he probably never will.  Now there is the slight chance that you never told him how you felt about it but just allowed it?  SPs are usually made to feel that their needs always have to go to the back, as in BM and bio-dad can noodle all they want "for the kids' sake," and if SM has a problem with it, that's on her.  But very few women want another woman, doesn't matter if it is BM or not, to be able to call her DH on over any time--putting SM in the wife #2 or GF #2 postion--unless they are entrenched in a polygamous compound.  That's all there is to it.  This is the US.  This is the year 2020 and not 1820. 

It's your BF.  He is the major problem.  He wants to play happy family with the ex and himself and kids.  Like before.  He knows you don't like it.  But three ex is coming first.  He not ready for a real relationship with you.

I am in a similar situation... or was... I think it's chilled out a bit after I told him to move out for a week and get his priorities straight. i sent myself so crazy I looked in his phone without permission. So not my style. 
My SO and his ex were just gross. On the skid swap they'd always meet at a pub and have a drink together, she messaged him constantly, not just about the skid. I found that she'd been sending him selfies would you believe! And my dense SO was  calling her by an old pet name Kit - short for kitten! She constantly texts his mum. She introduced his mum as her 'MIL' right in front of me, completely ignoring me, at my SD's birthday. They were never married. She invites herself to xmas. She always creates a kid issue when we are kid free to create attention for herself. There are no boundaries this woman won't cross. 
I am so fed up. SO and I are still together, and he finally seems to get why it hurts me and has put in some boundaries.  But to what extent? I don't know. I don't care anymore. There's  too much past hurt and I don't trust him. 
if it's really bothering you, and it should, tell him that it's not ok. It's hurts you and be absolutely prepared to bail him because of it. There is not reason they should behave like that. I have 2 daughters that I have 50/50. I barely speak to my ex unless it's about change of routine. I never send pics of the kids. And I don't expect them from him. 
Your SO is being lazy, having his cake and eating it too. Don't put up with it x