The essence of the drive through the Julian Alps is road 206, which crosses the Vrsic Pass between Kranjska Gora and Bovec further west in the mountain range. It's generally done as part of a loop around the Julian Alps, starting in Bled, going to Kranjska Gora, taking road 206 to Bovec and then continuing down the Soca Valley past Kobarid and Tolmin and returning to Bled through Podbrdo and Bohinjska Bistrica. Road 206 is the only road that truly traverses the Triglav National Park, it gets you closer to the interior parts of the Julian Alps than any other road. Take a look at Google Maps for reference, here's a link to the route I described: https://goo.gl/maps/Pshr3ZwGyyk
If you're looking for a shorter alternative in the vicinity of Bled, I'm afraid there's no road that offers the same kind of experience. However, a very good alternative that I recommend is going to the nearby Bohinj valley, already in the national park, visit the lake there, go to the Savica waterfall, take the cable car up to Mt Vogel with views of Mt Triglav, drive or even rent a bicycle and cycle into the upper part of Bohinj - Stara Fuzina, Studor, Srednja Vas, etc. It's a lovely, wilder and more rural region compared to Bled and it's very scenic. The near edge of the lake (Bohinj Jezero bus stop) is only about 30 minutes by car and 40 minutes by bus from Bled.