FBX is a game recorder so recording websites is not really its intended use. However, you can do it by selecting Fullscreen capture mode in the Capture section of FBX's setting, which will then simply record your whole screen.
Next time it happens, please use the Send a problem report option on the Support tab to send us your log files so we can check those.
The usual cause of 1s recordings is that the recording PC does not have enough system resources available when FBX tries to start recording so it is forced to immediately stop. The only other reasosn why recording should stop is that you are either out of disk space / have the option to stop recording when disk space is low enabled or you tab away from a fullscreen game.