You and your son are quite right to be frightened by this appalling junction. Many people do cycle on the pavement here because there are not even pedestrian crossings to walk across.
On the narrow section of the bridge it is useful to 'take the lane' as taught in level 3 cycle training. There is a rough ridge along the left lane going north which can be nasty in the wet.
At the North juntion there are 3 options depending on your final destination. If you can get to the ASL bike box at the lights then get a position away from the left where you are visible to all the drivers. If you cannot easily get into the ASL on the green light get behind a car and follow it through, do not try to weave between lanes through this junction.
Most parts of the city are best reached via the back streets off Minories. From the bridge junction go left but keep in the middle lane so you capture the lane aiming to be in the third lane going straight on at the next lights past station. At the next lights, after going under the railway, there is a pavement track going north on Minories to Crosswall which links to lots of lanes, many of which now allow to-way cycling on one-way streets.
If you are heading for North-East bits of the city then you can leave the bridge in the right hand lane, again in the ASL or following a car. Once in Mansell street there is a lane on the left, at the next lights go right then left onto another pavement track that takes you against the traffic flow to re-join Mansell st going north.
The third option is to get off and walk (or cycle illegally) the 60 yards from the bridge to the four stage pedestrian crossing towards Minories.
It is because of nasty roads like this that we recommend all cyclists, and especially men, should have level three cycle training to become experts in traffic.