In the radio settings page of the Meshtastic docs they describe the maximum speed of transfer for different setting combinations that have been tested. The fastest setting is the Short Range / Turbo, which can achieve up to 21.88 kbps: https://meshtastic.org/docs/overview/radio-settings/
The more common "LongFast" setting achieves 1.07 kbps. I just saved a few simple text-only HTML sites and they weight about ~2 - 5 kb, so it would certainly be possible to transmit them in a few seconds. There is some additional overhead with every packet, and in the EU at least we have a 10% duty cycle rule a 868 MHz, so only 6 seconds of every minute should be used for transmitting. Navigating through the page would be a bit cumbersome.
Since the html page will exceed the character limit from the Meshtastic app chat, it would probably make sense to make use of a dedicated app to send these kind of packets using LoRa instead of Meshtastic itself.
There are other methods of sending IP/TCP packets over packet radio. I am just now learning about this, and found this video yesterday: https://youtu.be/V0FAzMIsxMg?t=1413
With this technique you can use a radio to transmit packets to a Linux PC that behave just like an 'Ethernet' TCP packet. The video covers a related open source implementation called 'New Packet Radio' with which you can transfer 50 kbps - 500 kbps using radio packets in the 430 MHz band. These protocols work with frequencies and powers generally reserved for those with a HAM radio license: https://hackaday.io/project/164092-npr-new-packet-radio