if I already accepted a postdoc offer from Europe (only 1 year) and later received a tenure-track assistant professor offer from a research university in US,
I do not want to lose both opportunities, are there any good suggestions to handle these offers, how to write a letter to the professors/chairs?
Many universities would be willing to defer your start date by 1 year. A 1 year postdoc, with eg few teaching responsibilities and not worrying about finding your next job, can be extremely fruitful! So in that sense it also benefits your future university. It’s very reasonable and usual to ask for such a deferral.
EDIT: Regarding how to broach the topic with the TT/postdoc places, personally I would suggest to be up front and honest. Your mileage might vary depending on the university/institute. You said that the TT offer is from a research university. For any university or liberal arts college in the US where your TT job will have a substantial research component, this request should not surprise them. When you negotiate the terms of the TT offer, write something like the following:
Earlier this year I accepted a 1-year postdoc position at [[INSTITUTE]]. This would be a valuable opportunity to get my research up and running before starting a permanent position. Would it be possible to defer starting at [[UNIVERSITY]] for one year?
If you are nervous, you could consider being a bit more vague in the first part, for example by saying "I have the opportunity to go to [[INSTITUTE]] for up to one year", rather than saying you accepted an offer.
There may well be reasons that they cannot grant you a deferral, or maybe they can only defer for part of a year, but it's a reasonable to thing to ask for. It's worth remembering that once you have a TT offer, the ball is in your court, and it would be virtually unheard of for them to retract the offer.
If the TT offer is for a position with little or no research expectation, they are less likely to grant a deferral. But it might still be worth asking for one.
If they agree to defer, they will create a new offer with a new, agreed start date, and the procedure to accept will be the same one as now.
For the postdoc offer, again your mileage may vary depending on the location and your responsibilities. At my current (pure mathematics) institute there are numerous postdocs with zero teaching or administrative responsibilities, so if someone accepts an offer and then later changes their mind, that does not make a huge difference for us (yes this does periodically happen). This just opens up another spot for us to fill in a future application round, or off the waiting list. If your postdoc comes with teaching responsibilities, or you are supposed to be doing something specific in e.g. someone's lab, it will be harder for them to find a replacement. If you know someone at the postdoc institution, e.g. a postdoc mentor, it would be best to ask them directly for advice. If you were made an offer from someone's personal grant, backing out of an offer, while possible, is a delicate procedure and should be undertaken diplomatically.
Source: I work at a research institute in Europe where numerous people spend such 1-year postdocs (I am extremely curious if that is where your postdoc will be!)
I've run faculty searches in STEM at a top-5 University in the US. It is extremely common for us to hire someone in a tenure track (TT) position and then recommend that they do a one or two-year post-doc before coming to our department.
As somebody else stated, this usually benefits the department as well, because you learn something new, and they clearly think very highly of you already, which is why they gave you the TT offer in the first place. So just ask the department if it's okay.
