Another thing to add here is that we have very strange spelling rules. It sounds very funny when people try to enunciate the И in "ЖИТЬ". I take pity on those people when I hear that. I imagine myself saying it this way and I'm afraid that I'm going to break my tongue before I get it sound right.
The trick here is about the spelling rules of Russian. At school we're taught that "жи/ши" are written with an "И", not with an "Ы", as well as ЧУ and ЩУ are spelled with an У. But Ж & Ш are always hard*, they can't go along with soft vowels in speech, while Ч and Щ are always soft, they can't be friends with hard vowels. That's so weird!

I think it's absolutely physically impossible to say ЧАЩА. It's ЧЯЩЯ!
I think it's absolutely physically impossible to say ШИШКА. It's ШЫШКА!

If we take the word "широкий" for instance, it's pronounced шырокий. I don't know who invented this stupid rule to spell the ШЫ like that. Don't break your tongue by saying a Ш and then a soft vowel like an И or a Е. It's just absolutely inconvenient for native speakers of Russian.
Although I'm aware that it's ok to pronounce things like that for native speakers of English, because their vowels and consonants are not bound in pairs. But the natural way to say these things - is to pronunce ШИ as ШЫ (шип), ШЕ as ШЭ (шея), ЖИ as ЖЫ (жизнь), etc. The soft counterpart of Ш is Щ and it perfectly goes along with soft vowels, like щека (щека or щика), but it can't be followed by a hard vowel: щука (щюка -- take notice, the spelled У transforms into spoken Ю), and the Ч behaves in absolutely the same way, e.g. чудо should sound like чюдо.

So the thing to remember: don't not buy those ЧУ/ЩУ- and ЖИ/ШИ- things, keeping in mind that their spelling is quite illogical.

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*actually Ж can be soft, but I'm not used to the soft way of pronouncing the Ж sound, although I've heard it a lot.