Let's put this to bed once and for all. The sentence under discussion is:
> A woman in her forties sits on a bench, fixing the shrine with her gaze.
The construction in this sentence is perfectly standard in both British and American English, documented by reputable dictionaries, and in common usage across contexts from tabloids and young adult fantasy to newspapers of record and literary fiction.
Dictionaries: Several commenters have posted dictionary entries for related but distinct constructions like "fix a gaze on." Here are entries supporting the exact construction under discussion.
1. https://dictionary.cambridge.org/dictionary/english/fix-with
2. https://www.merriam-webster.com/dictionary/fix%20(someone)%2...
3. https://www.collinsdictionary.com/dictionary/english/fix, definition 9: "If you fix someone with a particular kind of expression, you look at them in that way."
- "He took her hand and fixed her with a look of deep concern. [VERB noun with noun]"
- "He fixed me with a lopsided grin. [VERB noun with noun]"
examples from other Collins entries:
- "The man fixed his interrogator with a steady gaze and spoke quietly but firmly." (https://www.collinsdictionary.com/sentences/english/gaze)
- "He pulls his other hand towards his face and fixes me with an intense gaze that has been well practised." (https://www.collinsdictionary.com/us/dictionary/english/inte...)
4. https://en.wiktionary.org/wiki/fix#Verb, definition 1.1: "(Of a piercing look) to direct at someone." (note that definition 1, but not 1.1, is marked as obsolete)
- "He fixed me with a sickly grin, and said, 'I told you it wouldn't work!'"
- "She sniffed, too, comprehendingly, and fixed her son with a relentless eye."
5. https://www.oxfordlearnersdictionaries.com/definition/englis..., idioms: "fix somebody with a look, stare, gaze, etc.: to look directly at somebody for a long time"
- "He fixed her with an angry stare."
6. Examples from other dictionaries exhibiting the construction:
- "To glare is to fix another with a hard, piercing stare" (https://ahdictionary.com/word/search.html?q=gaze)
- "Aron Nimzowitsch, a contemporary of Alekhine’s, would smoke a noxious cigar and fix his opponent with a dread stare." (https://www.dictionary.com/browse/nimzowitsch)