Can a computer be infected by malware through web browser?

Yes you can. Usually a proper Anti-Virus program will intercept these attempts though. Of course, browsers don't have "built in" backdoors/vulnerabilities but they can exist nevertheless. When such a vulnerability is found by a hacker or other malicious user, that vulnerability can be exploited to infect the visitor of the web page.

If something could be made 100% safe, there wouldn't be any need of Anti-Virus programs at all. There's always some creative solution that can be found to infect people anyway and such an exploit can only be patched after it has been discovered.

It's like typing an entire book on your keyboard, without checking for typos. You will only discover your errors when you start reading/reviewing whatever you typed. Reading in this case, would be to just "use" the browser.

Most exploits are in 3rd party plugins (such as Flash, PDF reader plugins, media, etc.) so the browser maintainers don't have full control over what is run in their browser. It's like having a house built by 4 different contractors who don't know exactly what the others are working on or how they are doing it...