Every time I want to download some app, download websites would lead to the Apple Store. Is it possible to download apps and install them in an iOS not through the Apple Store?
Generally no. However if you are technically adept you might want to look into jailbreaking your iOS device. Jailbreaking is officially off-topic here (so we can't help you) but there are a number of places online that help you to jailbreak your iPhone/iPad and install anything and change anything you want to– Steve ChambersDec 3, 2022 at 14:58
Not easily. By default, i-devices can only install software from Apple's App Store, after it has been approved by Apple. This is the so-called "walled garden", which some see as a good thing (increased security and oversight); and which others see a restriction of liberty.
As a result, there are (almost) no apps that exist to be downloaded from other sources.
It is possible for a developer to create an app and load it onto their own i-devices; it's also possible for organizations to configure iPhones and iPads to include their own software as part of MDM (mobile device management) software.
WRT creating & installing your own apps: What an interesting answer! When I read it I immediately wondered if the capability to load your own apps had led to the creation of a class of "open source" iOS applications that effectively break down the wall Apple has built? A quick search seems to suggest that has happened.– SeamusDec 3, 2022 at 23:47