To expect my 4 yo to walk next to me in public?

Reins are the answer. My daughter was a bolter until she was almost five (like me at her age).

There are the ridiculous people who liken them to dog leads but I'd rather my DC restrained slightly rather than dead in the road.

So you have a normal 4 year old then?

I feel your pain OP. Mine is also four and to give him his dues he will hold my hand next to roads and car parks etc because I’ve made it non negotiable, I told him it’s about being safe and if he doesn’t do it we must leave, he seems to accept that.
But in shops I do find it really stressful, he’s not even that bad either bless him, he just loves to explore and look at things, four year olds are inquisitive little creatures.
I think it helps if I tell him he can walk around but must stay close where I can see him, obviously sometimes the temptation is just too much I can almost see him thinking “Fuck it, it’s worth it” before bolting. He’ll grow out of it.

I would put reins on her.

I'd put reins on her. I would be clear... walk holding my hands or you'll go on reins. No ifs or buts. If I can't trust you to walk together and be safe then that's the consequence.

My son was like this at age 4. He absolutely wouldn’t hold hands and generally wanted to do his own thing in a very lively way. I had to watch him all the time and grab him if I thought he was about to do something dangerous. I used to look at other people and wonder how they got their kids to listen to them (also felt a bit judged TBH). Then I had my younger child who would happily walk next to me holding hands (so relaxing!) and realised kids just have different personalities, and crucially I was not a bad parent :)

3 kids, all went through a 'trying' phase when out and about. You need to keep reinforcing the expectation, a lot of patience as they try to establish their boundaries are needed. Consistency is all I've got really.

Love grocery delivery. My kids are almost 3 and 5. Either one of them with constant reminders might stay close and not touching stuff for the 3 seconds it takes to look well behaved but the whole thing is a nightmare. I make myself take them to low stakes shopping trips as practice and it kills me. I find it works better to tell them what to do (hands behind your back, stay close, come back) than to constantly shout no. But it’s still lots of shouting and coaching and all nonstop.

What are the consequences when she runs off and misbehaves?

I know four is young but I don't think it's so young that she can't understand how to behave in public. If nursery aged children can do it on trips to the park etc. then a four year old can.

Well mine are now young adults but they were also like this and if they wouldn’t hold hands we user reins. Oh and I would leave if they didn’t respond to warnings. And there was a reward eg the park or a small treat for listening to me. They are well adjusted men now. Reins are the easiest if she struggles with impulse control.

Sounds normal for her age. With that being said have you considered she may be dyspraxic? I was tested for it when I was in primary school and a major part is coordination, balance etc. I’ve grown a lot out of it now but I used to zig zag when walking because I couldn’t walk in a straight line and generally your daughter sounds like me! I am 26 now.

My 7 year old still doesn't walk next to me. Can't walk in a straight line, can't match pace, often lags behind or runs in front, picks up everything even though I've told him about a million times not to pick things up off the floor etc. I've accepted he probably won't do it until he's about 25 in all honesty

My Ds is just 5. I tend to hold his hand still when we are out walking somewhere like town centres.
He only walks ahead when no traffic like in parks or Small side streets where there no shops

This is completely normal, in my experience. My six year old usually runs ahead, it's like he can't help himself.

And yes, the others are right - don't take small children shopping. Once they're out of the pushchair it's impossible, they get bored and tired.

Mine didn’t.

She’s probably bored - shopping for a 4 year old is incredibly dull.

She’s really young - has she just turned four?
get back pack reins or just a accept your shopping will take longer

I would get something like a little life backpack to use until she learns to walk with you sensibly.

I think that is probably the default setting for most 4YOs. They can be very trying.