I ride a track bike setup with a single front brake on a route with small rolling steady hills and a handful where I have to really push to keep pace up.
I average 18-19mph on the track bike over 10 miles and 16 on the road bike on exactly the same route.
Also I let a lorry overtake me before a junction, I stay on his tail about 4 meters behind for a mile or two down a lane and overtake him before he turns off.
Got this as a result even with a backpack full of work gear, genuinely don't think I'd have been as confident doing it on a road bike.
Also if you do buy a fixie, weigh your gear ratio choice up on where you live, how long the route is, how many stop starts you're possibly gonna do etc. I'm on a 48-15 ratio and does the job nicely, climbed vicious hills on it quicker than road cyclists but it's knackering. You'll get fit quick if you stick at it because your body does have to work harder to ride one.