My site was experiencing the same issue and this is how I've resolved it...
If you'd like to continue using Jetpack, except for the Twitter portion, try the following and see if it works for you - its been the best solution for my WP sites so far:
first: deactivate the Twitter widget that comes with Jetpack and then activate the "Text" widget where you want your Twitter tweets to show;
second and third: log in to your twitter account and/or go directly to their widget page (https://twitter.com/about/resources/widgets/widget_profile) - (logging in allows Twitter to immediately identify your site's tweets and show them in the example while working on your color scheme) - choose your preferred widget style and set it up to complement your site, choosing colors and size - click on "Finish and Get Code" - then copy the entire code and paste it into the "text" widget you've added (in step one) into your sidebar (or whichever region) in your WP site, save and check out the live view. You should have exactly what you produced in the Twitter profile widget now displaying in your site with your most recent tweets!
This approach (so far) has allowed the use of the other Jetpack widgets to remain intact and maintain a "live" feed from Twitter.
Hope this helps!