How can I view my text messages online

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Use Messages in iCloud

Use Messages in iCloud – Apple Support (UK)

Access your messages on all your Apple devices

Access your messages on all your Apple devices - Apple Support (MY)

Use Messages on your iPhone, iPad, or iPod touch

Use Messages on your iPhone, iPad, or iPod touch - Apple Support

You can't view the Messages in iCloud.

Depending on your carrier, you may have options to view your SMS (but not iMessages) messages online. Contact your carrier to see if they offer such a service.