Its possible that the two problems are unrelated and happened coincidentally.

A few things I would check for the turn signals:

1. Do the 4-way flashers (hazards) work? (are the bulbs bad?)
2. Does the windshield wiper work?
3. Does the brights switch work? (with #2 - is the combination switch bad?)
4. Do the brake lights come on? (is the combination relay bad?)

About the temp display - does it display anything at all or just blank (or is the LCD completely black)?

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Daily Driver: 02 E430 4MATIC
In the family: '03 E500 // '04 ML500 // 64 220SE