> If there are problems at intersections, that means that the bike lanes aren't protected enough

Unless you design the intersections properly, then you're never going to achieve that protection. The best way to design them would be to follow how limited access highways/motorways interact with surface streets. In other words, they use interchanges.

> If there are problems at intersections, that means that the bike lanes aren't protected enough. Amsterdam for example doesn't have these kinds of issues because their design (and especially light signaling) is better.

I agree. The problem is that I have never seen those sorts of lanes in the US. The choice is between semi-protected lanes which may be worse or no lanes. I don't know precisely which is better, but I usually choose to not use the protected lanes due to issues at intersections.

We have intersections like this in NYC. Look at 8th and 9th Avenues in Chelsea for one example. There are separate phases for bikes going straight and vehicles turning left across them.

Seattle also has pretty good signage for their (very limited) protected bike lanes - typically a green-painted turning lane, with yield markings and signs. I still see drivers ignoring them from time to time - I think Mad-Eye Moody's mantra is a good one to keep in mind when riding a bike: "Constant Vigilance!".

We also have a few of these kinds of lanes in Austin. There are a few downtown and a few in East Austin.

Where specifically? I'm not aware of bike lanes that have a special light for vehicles turning across the bike lane in Austin.