Which free online academic plagiarism checker is closest to Turnitin? Which free online academic plagiarism checker is closest to Turnitin by quality  standard of rigour?

Turnitin features that even somebody plagiarize the context of a passage by use synonyms or simply rephrase the sentence, it is still able to catch the student cheating...

Please notice that Turnitin is not free and some of free software are not rigorous enough or even people called it a joke!
I have recently (2 year ago) done a fairly thorough search for good, free alternatives to TurnItIn. My motivation was that my school did not use TurnItIn but that does not stop me from caring about my students' work. In the end, I found all of them (and there are many) are simply terrible in comparison to TurnItIn.

I've been using TurnItIn for 2 years now and I find it quite solid (of course there are things I would change but the basic functionality works well). The system has flaws but as I see, they maintain it and when a vulnerability is found (allowing a student to game the system), they generally fix it.

Add to this the fact that most often (my) students copy from previous students, searching freely available online sources will not do a proper job.

TurnItIn has a bit of a natural monopoly so it seems unlikely that any free alternative will ever gain much market share, meaning they will not have the papers to check leading to a vicious circle.

I see you are a high school student which makes me wonder why you want to know. Without knowing your true intention, I'm afraid my answer will be less useful to you.

If your purpose is to recommend a system for your school, they need TurnItIn. If your purpose is to check your own papers, you should use TurnItIn's student version (I know, it's not free).