Is it polite to say "I urge you to..." or "I request you to..."

The first is fine; the second is overly formal to the point of awkwardness. Use these:

I urge you to choose Chicago.
Please give me an update by this Friday.

Thanks, Mister Micawber.

I was a bit surprised when I heard my project manager say "I request you to..." I will let him know the use of "request" is not correct here.

The first is ok but as a native English speaker I'd never use the second one.

Also, the first one is only used in a more formal way, when you want to emphasise that it is important. We use this when we are trying to convince someone to do something that they may not want to do. I.e, "I urge you to reconsider". There are better ways to just recommend something. Hope that helps.

I don't think it's very wise to correct your manager, Jingtian. Managers don't handle correction from subordinates very well.

I request you

Thank you so much it was helpful for me.