In formal writing about government, each branch has a set name you shouldn't paraphrase. The legislative branch makes laws, the executive branch enforces them, and the judicial branch interprets them. Say "legislative branch," not "law-making branch." These are the standard terms in US government and in writing that compares countries; British reporting more often says Parliament, the Government and the judiciary.
The three branches of government
legislative branch /ˈlɛdʒɪsleɪtɪv bræntʃ/ — The legislative branch passed the bill after months of debate.
executive branch /ɪɡˈzɛkjətɪv bræntʃ/ — Enforcing the new rules falls to the executive branch, not the courts.
judicial branch /dʒuˈdɪʃəl bræntʃ/ — The judicial branch decides whether a law is constitutional.
Not ""The law-making branch makes laws." (not the standard term)" — say ""The legislative branch makes laws." (fixed political term)". The formal name is legislative branch. "Law-making branch" is understandable but reads as a paraphrase, not the real term.
Try it: Which branch is responsible for enforcing laws? (the executive branch / the legislative branch / the judicial branch) Answer: the executive branch — The executive branch carries out and enforces laws. The legislative branch makes them in the first place, and the judicial branch interprets them and rules on whether they hold up, so neither fits "enforcing."
Political processes in fixed phrases
A lot of political vocabulary lives in set collocations. Voter turnout is the term for the share of people who actually vote, voter participation is a real term too, but it is broader and doesn't name that figure. A grassroots movement (not "basic movement") is driven by ordinary citizens. Gross misconduct (not "big misconduct") is the term from employment rules for wrongdoing serious enough to justify dismissal.
voter turnout /ˈvoʊtər ˈtɜrnaʊt/ — Voter turnout was unusually high in the last election.
grassroots movement /ˈɡræsruts ˈmuvmənt/ — The campaign started as a small grassroots movement in a few towns.
gross misconduct /ɡroʊs ˌmɪsˈkɑndʌkt/ — The minister resigned after being accused of gross misconduct.
Not ""Voter attendance was high this year." (not the term for this)" — say ""Voter turnout was high this year." (standard political term)". The share of people who actually vote is voter turnout, the term news and academic writing use for that figure. (Voter participation is a real term, but broader.)
Try it: A senior official is dismissed for serious wrongdoing. Which phrase fits? (gross misconduct / big misconduct / grave participation) Answer: gross misconduct — "Gross misconduct" is the fixed phrase for serious wrongdoing that justifies dismissal. "Big misconduct" uses the wrong intensifier, and "grave participation" isn't an English collocation at all and doesn't even mean wrongdoing.
Try it: A campaign built and led by ordinary citizens rather than officials is best described as a ___. (grassroots movement / basic movement / ground-level party) Answer: grassroots movement — "Grassroots movement" is the set phrase for action led from the bottom up by ordinary people. "Basic movement" isn't a real collocation, and "ground-level party" mixes up a party (a formal organization) with a movement and still isn't idiomatic.