That link you posted almost completely contradicts you. Here are a few choice bits:
> triclosan’s ability to behave as an estrogen antagonist also suggests that its presence in the body alongside estradiol may actually lower risk for cancer development [22,23]. However, whether triclosan raises or lowers risk of cancer through estrogen-related pathways, and possible effect modification by estradiol, have not been examined in human studies.
> Together, the results of these in vitro studies suggest that FAS inhibitors, such as triclosan, may be an effective inhibitor of cancer growth, though no study has examined whether triclosan reduces cancer growth or incidence in humans, or what co-factors may potentiate any effect.
> the results of animal studies to date have been mixed, showing null, inverse, and positive associations.
> Both short and long-term studies have examined triclosan following oral and dermal exposure, showing rapid excretion from the body and no evidence of toxicity, irritation, or thyroid hormone disruption
> [studies] have not addressed experimental findings that suggest triclosan’s estrogenicity may function to either stimulate or inhibit human estrogen-dependent cancer cell growth.
Basically, a bit of speculation regarding triclocan's estrogenicity, one positive animal result with apparently no human application, and a handful of weak evidence showing that it just might actually decrease cancers.
Detergent formulations, such as for dishes and laundry, are simple and entirely uncontroversial with regard to health and safety.
Except for parabens, I'm unaware of any ingredients in cosmetic soaps or shampoos that are plausibly suspected of even faint carcinogenic activity. I'm happy to consider any contradictory evidence you may have.
> Many of them will also have a number in the hazmat rhombus, which is also not a good thing.
That doesn't mean they're carcinogenic, or even that they're especially harmful.
Even if by "a number in the hazmat rhombus" you specifically mean a number in the health square of the NFPA 704 [1] diamond, the health hazard of a substance is assessed based on the danger of exposure to a concentrated, pure form of that substance. Acetic acid rates a 3 out of 4 on that scale, for instance, because glacial acetic acid is quite corrosive -- but that doesn't mean that vinegar is dangerous.
Cosmetics are a bit of a black box, but what carcinogens are supposed to appear in soaps, shampoos, detergents, or most common cleaners?