Move your bowels may be too polite and sounds strained, and merely saying shit sounds offensive. What do native English speakers say then when you need to move your bowels, especially when a parent asks his toddlers whether they need to shit?
I'll assume you're talking about usage in mixed company where good manners are required. North Americans seem to use 'go to the bathroom' to cover a wide range of excretory and ablutionary functions, regardless of whether an actual bathroom is within walking distance. Brits might use the similar 'go to the lavatory' or 'go to the loo'. We Australians will typically say 'go to the toilet' or 'use the toilet' - 'toilet' being a word considered almost taboo in certain other English-speaking cultures - although 'bathroom' is creeping into common usage.
If you're talking specifically about bowel movements as opposed to urination, I wouldn't think there is a need to specify one or the other in polite circles. If with close friends or family, you might say something like 'do a poo' or 'take a crap'. There would be a thousand synonyms for 'having a shit' I'm sure.
In AmE, 'Poop' is the current terminology. That is, in a squeaky voice:
I have to poop
is what the kids say these days. Adults usually don't tend to feel like sharing the particular mode of ...export. They might euphemize even further and just say 'I have to go' or 'I really have to go'.
"Number one" for urinate and "number two" for defecate was the terminology I remember from childhood (but is also not childish and not taboo (that is, not like 'shit', it is very plain sounding))
As you can see from the answers so far, this is not a simple question to answer.
We've had answers that specifically mention Australian and US usage. I'll do my best to answer for Britain although I'm afraid there will be differences in different parts of the country.
In the version of British English that I speak I have known variants change with the generations. I have also known differences between different families of the same generation.
Young child: Mummy, I need a wee! (urination)
Young child: Mummy, I need a poo! (defecation)
Mother: Do you need a wee/poo?
Farmer: John, can you clear up the cow shit from the yard? (I have worked on a farm so I know this is used without there being any sense of it being a bad word.))
Male friends at a pub
Man A: Where's Bill?
Man B: I think he's gone for a crap." (possibly 'shit')
Women friends at a pub
[I don't know what women say when men aren't around]
"Where's Jane?" "She's in the Ladies."
"Where's Bill?" "He's in the Gents/Mens."
I'll add others if requested.
It's sad to see that conventions of "culture" seem to ban the casual and acceptable mention of normal physical functions and deprive us from using commonly accepted terms to describe what we all do frequently. A medical doctor once told men that "normal" bowel movement frequency is anything from several times a day to several times a week.
It's perfectly normal to inhale. We all do it. We all also exhale. It's not only normal, it's actually required to stay alive. The same goes for food and drink. It's perfectly normal to eat. Or to drink. But when it comes to using proper words for the "exhale" equivalent of eating and drinking we're struggling to find words that are not offensive or vulgar.
We feel the need to disguise the "exhaling" equivalent with words that don't quite spell out the real thing, so as not to upset people or breach agreed conventions of acceptable language. We use "wee" which literally means "little" in English, or we prioritise the bodily activities with "number one" and "number two". (Who did ever come up with that? )
Words that describe the actual bodily activities are either euphemistic and rather ancient sounding phrases like "move your bowels" or more often than not Latin phrases which have made it into the common language by way of medical terminology, like "urinate" or "defecate".
Sadly, there are no commonly accepted "standard" terms to describe "passing urine" or "moving your bowels" in plain English with language that is not offensive or upsetting to one party or another.
Depending on the region, local customs and other factors like education, etc., it may be perfectly allright for a parent to ask her toddler if she's had a "shit". In other regions or other social circumstances, the parent will be more inclined to use "have you done a poo-poo?" as the defining term.
I've heard families use the word "stinker", as in "Have you done a S?" [sorry]
English is spoken as a native language on several continents, with quite different rules around socially acceptable terms.
The very intimate parent-to-child communication about bodily functions can probably not be defined with a single rule for all English-speaking communities world-wide. There will be many different word choices, depending on the country, custom, value framework, religion, rank in the social model, etc. that define what the locally acceptable term will be.
Therefore, a "common expression" does not exist. It is very much dependent on the socially acceptable terms of the immediate environment.
A nanny caring for children of the English Royal family will use words that differ quite a lot from those used by a 3rd generation beneficiary in NZ, or those used by a white collar worker in Kentucky, US.
Therefore, I think that there is no "common expression" for all English-speaking communities, especially if the focus is on parent / toddler communication.