The astrophysics journal (ApJ) is a really good journal. A majority of good astronomy papers are published in ApJ or in MNRAS. Big things go in Nature/Science + then there are other smaller journals.

(At least this is what I tend to find. This may just reflect my biases - US based, in the cosmology field.)

But just because something gets published doesn't mean it is right :) I think non-scientists don't know what "peer review" actually entails! First, as this was a MOND paper, it could well have been reviewed by someone who favours MOND. Second, even if the reviewer doesn't favour MOND, if the steps taken and the arguments given seem reasonable, I expect they would suggest it should be published.

I have no issue with the paper being published. It is important that theories have advocates who put forward the best argument for them. What I do take issue with is the skewed presentation in popular science. What sells is exciting and new, not slow and steady. And 99% of science is slow and steady.

Also, peer review in astronomy generally consists of correspondence with a single reviewer. There may be multiple rounds of review but a significant majority of papers submitted by professional astronomers doing serious research to journals are ultimately published. It is not the norm in astronomy to be rejected outright from a top journal and then submit to a lower tier journal, instead almost everyone publishes in a few popular journals. This is a tradeoff, but definitely in astronomy at least this means some iffy papers are ultimately published, even in ApJ, and you must always be aware when reading the literature that the paper you are reading is not infallibly correct just because it was published in a journal.

ApJ is one of the most respected journals in astronomy.