How do I actually improve teaching across my school?

You can manage the building, but actually improving teaching across it eludes you, and that is the deeper, harder work. Managing keeps things running; improving teaching means changing what happens behind many closed doors, in the hardest-to-reach place, another person's practice. It is normal that this feels different and harder than everything else you handle well. Teaching improves through trust and focus, not directives from the office. You cannot fix it all at once or from a distance. It changes when teachers feel supported rather than judged, and when everyone works on one clear thing together rather than ten scattered ones. Depth beats breadth. Pick a single meaningful improvement and go deep across the school. Choose one instructional focus that would matter most, and put your energy there for a real stretch of time, in classrooms, not just meetings. Be present, supportive, and consistent about that one thing. Improving teaching is slow, close work. You do it by narrowing your aim and staying with it long enough for it to take root.