Certain websites devoted to idioms claim that they--idioms-- are not used in formal conversations or writing; that is, they claim that these creatures are always used in only informal situations. As the site is a British website, is this a common belief among the British, or is this a mere hiccup in the website? For "according to" and "at first glance" are far from "informal".
One problem is that the term idiom has various definitions. I would be surprised if the websites you refer to make the claim that a.) according to and at first glance are idioms and b.) such idioms are not used in formal conversations or writing. This is simply not true.
A narrower definition of idiom is that an idiom is a phrase whose meaning cannot be deduced from its constituent words. Examples are:
She's having kittens. = She feeling nervous.
You get up my nose. = You irritate me.
It is true that informal language abounds with idioms in this sense of the word. It is also true that you would not expect to see some of these idioms in formal conversations or writing. It is not true, however, to claim that no idioms are found in such situations.
Here are just three examples from the Telegraph, a serious British newspaper:
The latest can of worms to burst open is the notion that an independent Scotland should have a properly funded pension scheme.
Iran decides to give its Western adversaries a taste of their own medicine.
Disagreements within the Conservative Party about Europe dominate the headlines this morning, but it's a storm in a teacup compared to Labour's deep ideological divisions.