Formal pairs suit Writing Task 2. Neutral ones work anywhere. Informal ones belong in Speaking.
1.commit a crime
neutraldo something illegal
Young people who commit a crime for the first time need guidance, not just punishment.
Commit also goes with offence and fraud.
Avoid: do a crime → commit a crime
2.the crime rate
neutralthe number of crimes in an area over a period
The crime rate has fallen in cities that invested in youth centres.
Rise, fall, reduce and a sharp increase in collocate with it.
3.serve a prison sentence
neutralspend the time in prison that a court has ordered
Prisoners who serve a long sentence often find it hard to find work afterwards.
Serve is the natural verb. Do time is slang.
4.impose harsh penalties
formalofficially give strict punishments
Some governments impose harsh penalties for drug offences.
Impose a fine and impose a ban follow the same pattern.
5.act as a deterrent
formalmake people less likely to do something
Supporters claim that long sentences act as a deterrent to would-be criminals.
Serve as a deterrent is also correct.
6.break the law
neutraldo something that is against the law
Drivers who break the law by using a phone should lose their licence.
Obey and respect the law are the opposites.
7.law-abiding citizens
formalpeople who obey the law
Surveillance cameras make most law-abiding citizens feel safer.
Keep the hyphen.
8.repeat offenders
formalpeople who commit crimes again after being punished
Education in prison reduces the number of repeat offenders.
Reoffend is the verb; reoffending rates is the statistic.
9.rehabilitate offenders
formalhelp criminals return to normal life and stop offending
Prisons should aim to rehabilitate offenders, not simply punish them.
Rehabilitation is the noun, often contrasted with punishment.
10.petty crime
neutralminor crime such as shoplifting or vandalism
Petty crime is often linked to boredom and a lack of opportunities for teenagers.
Serious crime and violent crime are the contrasts.
11.abolish the death penalty
formalofficially end the use of execution as a punishment
More than two-thirds of countries have abolished the death penalty in law or practice.
Capital punishment is a formal synonym.
12.enforce the law
formalmake sure that people obey the law
The police cannot enforce the law effectively without public trust.
Law enforcement is the noun phrase.
13.tackle the root causes of crime
formaldeal with the basic reasons why crime happens
Longer sentences do little to tackle the root causes of crime such as poverty.
Address is a formal alternative to tackle.
14.community service
neutralunpaid work that an offender is ordered to do instead of prison
First-time offenders could be sentenced to community service rather than jail.
Sentence someone to and carry out collocate with it.
15.crack down on
neutraldeal with a type of crime much more strictly
The city council cracked down on illegal parking last year.
A crackdown is the noun. Common in news writing and acceptable in essays.