If you belong to such a category that the mere belonging to it is a death sentence, if revealed, the situation is vastly different. You have to act more like a secret agent or a spy. This means constant, pervasive, fastidious opsec. Any death-sentence-invoking activities should be strictly separated from the normal civil life. Only use the normal browser to visit commerce, official news, and government web sites. Everything that is not openly pious and loyal should belong to ephemeral VMs with a fresh browser install every time (preferably several different), VPNs that are indistinguishable from legitimate web traffic, like XRay, truecrypt-protected media with some plausible deniability data, etc. It all takes quite some technical chops, but is not sufficient. Many other small details, related to technology or not, have to be carefully, well, sanitized, and any small slip can out you.
Such undercover life, while possible, is very tiring, takes a lot of extra time and energy, and noticing this also may mark you as suspicious.
Another browser API that may slightly help track you is a minor problem on this background, unless it pierces any of your layers of protection.