Secure Android browser
Chrome for Android definitely does (http://www.theverge.com/2015/12/7/9865928/google-chrome-safe-browsing-android-malware). The safe browsing feature isn't yet supported by Android WebView so any WebView-based browser (e.g. CM browser, Ghostery, or Lighting Browser) lacks it.

Firefox on Android should support safe browsing as well, as per their blog posts: https://blog.mozilla.org/futurereleases/2012/11/26/firefox-for-android-beta-improves-phishing-and-malware-protection/ and http://www.morbo.org/2012/10/phishing-protection-on-mobile.html. Not sure why doesn't it work for you.