Britain's crime minister has bag stolen at police conference

Grammatically, introducing "her" doesn't reduce the ambiguity.

The sentence still could refer to her ordering someone to steal her handbag. E.g. "the crime minister had her partner killed" will almost always refer to a murder for hire.

"The crime minister's bag was stolen" is less ambiguous. But it's English, everything is going to be a bit vague.

I agree, the headline is unclear. It should be "gets her bag stolen" or as you suggested, "had her bag stolen".

Better yet, the main object of this sentence, her bag, should be at the front of the headline. "Bag stolen from Britain's Crime Minister" is funnier (in my opinion), more concise, and eliminates the "where", which can be put in a byline/sub-headline, or just left in the article as a gem for the reader to find.

That's not how I first understood it, but I like your version better. Maybe she had it stolen as a demonstration of crime.

That's English for you, it's often incredibly hard to express yourself without introducing ambiguity.

That's the reason "legalese" is a thing thing. When you really need to avoid ambiguous wording the language changes quite a bit.

that or the minister has a bag, that was stolen from someone at a conference earlier.