Is water from the bathroom sink any dirtier than water from the kitchen sink

Assuming there are no other drain problems occurring (because you did not mention any) you most certainly have an obstruction of the branch drain line leading to that sink.

Option 1- run a good quality drain auger down the drain, through the trap, and down the branch line as far as it will go.

Option 2- take the trap back apart and run the cable directly down the drain line.

For either option, it is important that you use the correct technique and persevere until the length of the cable has gone down. It's not easy, be prepared to work. There are numerous instructional videos on-line, this one shows the correct technique but the person is using a cheap sub-standard auger.

I always cable from the top and run water until it backs up a bit, so I can tell when I have hit the clog (because the water suddenly drains). I highly recommend that you invest in a good professional quality auger, I did and it has saved me hundreds and hundreds of dollars over the years:
Always wipe off the cable with a rag as you remove it from the drain, and always spray it down with your favorite water displacing spray lube that comes in a blue and yellow can or it will rust in the drum.

Since the hot water's flow rate depends on keeping the boiler pressurized, and this can only happen by feeding cold water to the boiler as you draw hot water, the total flow rate is limited by the flow rate of the cold water feed to the house. You will see a similar reduction in flow rate if you open both the bathtub and the bathroom sink's hot (or cold) faucets, or any combination of separate water valves in the house.