What actions can I take to make web browsing on mobile phone (android) as safe as possible? My main concern are sites that would install malware (has happened before on desktop computer). On a computer, I would run browser in a sandbox environment (e.g. Sandboxie). Are there similar alternatives for mobile phones? Or are mobile environments more completely sandboxed by nature?
Installing malware through websites is not as common as you seem to think. The malicious website needs to exploit a vulnerability in your browser which permits to get root access (quite uncommon).
So, as far as your browser is up to date, only the 0days (rare) could affect you.
But, even with less privilege, the malicious website could still tamper the behavior of your browser (spying you, insert scripts in the visited pages, ...).
Sandboxes can be a partial answer to this threat because they focus on isolating a process from the rest of the system (to avoid a malware spraying everywhere). They do are present in Android but they are not the ultimate protection.
But many websites will be broken as client side scripting is now a huge part of the Web 2.0.
Security experts, who want to keep their computer safe, rely on up to date Linux (android is Linux-based) and avoiding crap websites with malvertising (like porn, download, buzznews, ...).
It is less overkill than disabling the scripts.
Maybe take a look at the brave browser which is designed to limit the websites capabilities on your browser.
Android has a lot of security features like SELinux, permission system and sandboxing already to try to prevent exploits and malware.
But as with a lot of security, the user might be the weakest link. It is highly unlikely that a malicious website will take root access or even just exploit your browser without you noticing, but what is much more likely is some shady website downloading a malicious .apk on your phone and tricking you to execute it.
The solution to such attack is disabling installing third-party apps, which is already disabled by default. Also most browsers will warn you against downloading an .apk, so at this point getting infected by such malware would be like willingly jumping into a cage with tiger despite all warnings.
The bigger issue to Android security can be apps from Google Play, which not always are closely inspected and examined and can contain malicious code, so you should only ever install programs that you trust.