Carrying out: tiến hành / thực hiện + verb

Tiến hành (carry out, conduct) and thực hiện (carry out, implement) put a formal frame round an action. They go before another verb or a noun for an activity: Cảnh sát đã tiến hành kiểm tra. The police carried out an inspection. Literally that's "proceeded to inspect"; the meaning is simply "inspected", put in an official way.

Tiến hành is about starting and conducting an activity, often one with several steps: tiến hành điều tra (conduct an investigation), tiến hành khảo sát (carry out a survey), tiến hành phẫu thuật (perform surgery). In the methods section of a paper it's standard: Nghiên cứu được tiến hành tại ba tỉnh. The study was conducted in three provinces.

Thực hiện is about carrying through something planned or required: a plan, a policy, a project, a duty, a procedure. Thực hiện dự án, thực hiện chính sách, thực hiện nghĩa vụ. It's also the word for "made by" in news credits: Bài viết do nhóm phóng viên thực hiện. Report by our team of reporters.

Both often appear with được for a passive, and with do to name who did it: do Viện Dinh dưỡng thực hiện (carried out by the Nutrition Institute).

You meet them in news, police and parliament reports, and papers. In conversation, drop them and use the plain verb: Công an kiểm tra rồi. The police have checked. In a chat, tiến hành sounds like a press release.