I already know what the problem is, you're getting too caught up trying to take down one guy that you're letting everyone else drive away for free.. What I do is I always try to catch the people upfront. While i do this i bump into the other racers if opportunity knocks. You have spike strips to catch the guys behind you. It's particularly useful to use them when they decide to emp you. Spike strips aren't hard to dodge, if they happen to release them right in front of you, then tough luck.
Plus when you're behind in hot pursuit, the cops get so much catchup boost that it's silly. Should be no reason why you can't catch up considering the fact you're practically driving at top speed the whole time. But for the most part try not to get boost happy as you don't get a lot of boost unless you're doing a lot, save it for when you get that golden opportunity to deal a massive blow. Most events are done a lot faster through melee attacks I find, and spikes, EMPs and roadblocks help finish them off.
The best ways to score lots of damage if you're up ahead is to call a roadblock, before you pass through the opening, drop a spike strip. Time it so that by the time they reach the roadblock, the strips would have been placed in that gap. They'll get spiked or will have to blast through the roadblock vehicles, doing damage either way.
Alternatively, if you're behind a bunch of cars, call in the roadblock. One guy is bound to mess up causing a potential pile up, when these guys are tumbling in their wrecked state, smash into them, free kill. this only works offline.
This method is particularly useful in interceptor, push the racer away from the gap enough that you know they won't make it through the roadblock. slow down, they crash, move in for the kill. if they wrecked quickly stop beside the wrecked car (as close as you can without moving too much), busted. if they don't wreck, rear end them for damage.)w
If you like racing the most, like me, then just play that, you're not required to play cop events. I'd play em to unlock all the cops to play online however...
tl;dr Don't go for the first racer you see, side check him or whatever and keep moving up until you're at least in the middle of the pack. Don't get hasty with your boost usage. Don't tailgate racers for too long if that event has spike strips... only tailgate when they just used it (because they require charging time).