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Tower Bridge apparently has a clearance of 8.5 m at high tide which isn't bonkers high. You could probably make an Olympic sport out of jumping from that sort of height but you wouldn't catch me doing it. Even at low tide 16m is the kind of height (other) people jump when doing things like coasteering.
The obstacles bit is about some local knowledge or a guide again. I took my daughters coasteering in Cornwall this summer and we paid a reasonable fee per person to be in a small group with a guide that knew the area. Others were on their own and our guide commented that where some people were jumping would be too shallow to land safely in about an hour, hopefully they knew that.
Fortunately for my coasteering exploits my daughters are less adventurous than me so I can pretend to be invincible when really I was bricking it at the though of going much higher...
Thanks. Other things the latest article mentioned were specific to jumping from height, which I'm sure is not something that even Thames swimming advocates would recommend:
I'm sure that different water levels can be a problem even when not jumping, or indeed objects underwater (such as those kl mentions exist in Shadwell Lido, but which in this case again would probably mainly affect people jumping in).