4 Punctures in 3 Weeks

This post is a plea for advice, for help in my fight against flat tyres.
I recently switched from using a mountain bike with 26″ slick tyres to using a road bike with 700×25 tyres for my daily commute.
My new bike, a Genesis Aether 10, which is my first road bike, came fitted with Continental Ultra Race tyres. They seemed fast and lively at first, but I’m beginning to doubt if they’re up to the challenge of Cardiff’s streets.
In the past 3 weeks, since I switched bikes, I’ve had 4 punctures. Each time I’ve found a tiny shard of glass embedded in the tyre. I was riding on the same roads as I had ridden my old bike, for 10 months, without a single puncture.
Is this a common experience when switching? What’s the best solution?