The TurnLanes-tagging plugin works on both uni-directional and bi-directional roads. You can add lane counts as well as individual turn lanes in each direction. The plugin intuitively provides presets for the most common, and recently used, turn lane combinations, and also automatically updates lane count based on the number of turn lanes for the road.
To get started, activate the plugin from JOSM Preferences > Plugins > turnlanes-tagging and restart JOSM (if needed).
To use the plugin, select a road and open the turnlanes-tagging plugin in the edit menu. Choose the type of way (uni- or bi-directional), then set the number of lane. Click the icons to specify the allowed direction of movement per each lane.