And then you get roundabouts with traffic lights like in the UK. I'm sure there are good reasons for that, but, as someone that hardly drives in the UK, I was very very surprised the first time I encountered them.
If you imagine traffic on a motorway going North/South and an A-road running East-West, at busy times, the traffic going from East to West and West to East will be significant enough that traffic coming off the motorway is not able to find enough space to enter the roundabout (or, not enough to meet the demand). Without traffic lights, traffic will soon back up down the slip road onto the motorway, or worse, will try to force its way onto the roundabout instead of giving way to the right.
Traffic lights make this much safer.
In general the UK has very few flyovers. I sometimes suspect the SF Bay Area has more flyovers than the entire of the UK.
https://www.google.com/maps/@40.7957351,-74.3405952,17z
It's a circle, with traffic light, but also still an intersection. Fun!
Traffic lights are also used to give priority to pedestrians and cyclists on junctions where it would be unsafe to rely on zebra crossings or pedestrian priority - think sharp and low visibility corners.
As far as roundabouts go, the UK has been replacing big multi-lane roundabouts in areas with lots of pedestrians and cyclists with two-way-traffic horse shoes (e.g. Old Street and E&C, plus others outside of London). This allows the centre of the roundabout to be used as public space, and has been found to be safer for vulnerable road users.
I'm all for light-free junctions where possible, and light-free roundabouts work REALLY well in low/medium traffic. But you can't just abolish traffic lights - unless we also abolish cars at the same time.
So taking the 3rd exit on a bike either means following the road all the way around, or just going directly from the entrance to the 3rd exit, as cycle lane is usually preserved where the road used to be.
The safest route for bikes is to take the segregated bike route at the bottom of the horseshoe, then using the mixed bike and pedestrian crossing to go directly over the mixed used paved area, then use the mixed crossing on the other side to join the A201. Either that, or use the segregated cycle path, then horseshoe near London Rd.
But honestly, on a bike you would just avoid the horseshoe completely. There’s usually better cycle routes that are more direct, and avoid large junctions like the E&C horseshoe.
The essence of of is segregated cycle lanes with cycle-only traffic lights. Nothing magic.
Neither of them are fully separated (yet at least - they're still being worked on), but they're a lot better than what came before.
Trafalgar Square also used to be a roundabout back in the day, though it got converted to a peninsula before bicycles were fashionable so that's not a great example.
Highbury Corner is another example you can look at.