How many messages does your Inbox folder contain?

I have 4,3 GB mails in my inbox.

4.3 GB is very big for an Inbox folder. Inbox folders are not meant to store decades of message traffic and it's better to store messages you want (or you need) to keep in Archive folders. You may also create thematic folders in the Local Folders for long time storing. You may try to rebuild the Global Database https://support.mozilla.org/en-US/kb/rebuilding-global-database Rebuilding the global-messages-db.sqlite file after restart of Thunderbird might take several hours depending on the number of messages in your Thunderbird profile.

1. Rebuilding the Global Database did not hjælp. 2. How I can Archive bedst my messeges ?

You may archive messages using Thunderbird's "archive" feature, following explanations in this article https://support.mozilla.org/en-US/kb/archived-messages

Instead of using the archive feature many Thunderbird users create thematic folders in Local Folders to store messages from their Inbox, such as Banking, Invoices, Family, Taxes, ... etc

Remember: after archiving your Inbox messages, whatever the method you'll use, you have to "compact" the Inbox folder: otherwise the Inbox folder will never get smaller https://support.mozilla.org/en-US/kb/compacting-folders

@Adrian, I just remembered that if your email account is set up to run with the POP protocol and if the Inbox folder has very many sub-folders your issue might be caused by https://bugzilla.mozilla.org/show_bug.cgi?id=1979580 "'Search Messages' on many (hundreds of) local folders hangs."