Hopefully the whatever changed with the site will be resolved. Meantime, a few unsatisfactory workarounds.
Turning off javascript seems to work a bit. But that is something you may not like doing every time you need to make a selection or change a station.
Pressing the choice you want, say the 'Location' one, and then sweeping down from the top and refreshing by tapping the circular arrow. If the arrow isn't there then tapping the name bar so that it turns blue then tapping 'Go' on the keyboard that's popped up will refresh. Problem is you'll need to do this for most every selection made.
You might try
TuneIn Radio: Find and listen to Talk, Music and Local Radio Online instead of the regular Tunein.com site. It's faster with a different layout but doesn't seem to have all that is offered through the regular site. And it sometimes requires you to turn the tablet to portrait to see all that it does display.
Last might be to go to Bing and type in Tunein. Save the resulting page as a favorite. As per the attachment, Bing has that nice feature of letting you search the site's contents directly from there. Once in, you may need to use the sweep down and refresh at times, though.