The chief executive of the Electoral Commission has told MPs that fines for breaches of election rules should be increased from their current level of £20,000 per offence as part of a wide-ranging drive to reform of laws that have become “old and fragmented”.

Claire Bassett told the digital, culture, media and sport select committee that the existing structure of fines meant breaches of the rules were regarded by political actors as “a cost of doing business”.

Concern about the state of election law grew after the 2016 EU referendum, and the commission is continuing to investigate whether Vote Leave breached its £7m spending limit by colluding with another pro-Brexit campaign in the weeks before the vote.

“I don’t think it is any great secret that our electoral law is old and fragmented. It is something that has developed over the years, and right across the work that we do we struggle with the complexity that creates,” Bassett said. However, she added: “I wouldn’t go so far as to say it is not fit for purpose.”

When asked by the committee chair, Damian Collins, to spell out specific examples, Bassett said: “We have repeatedly said that our maximum fine of £20,000 [per offence] is too low.” She did not spell out at what level she thought the maximum should be set. “That is for parliament to decide,” she said.

Bassett also said parliament should consider extending the time period during which the spending of groups active in referendum campaigns is regulated. She said the existing window “creates challenges in a referendum setting” because groups could spend money in advance that could not easily be monitored.

A week ago it emerged that Leave.EU received more than £12m of campaign services from a company controlled by the donor Arron Banks prior to spending limits taking effect in April 2016. Leave.EU’s referendum spending cap during the regulated period was £700,000.

Electoral Commission executives said investigations into spending by the Vote Leave and Leave.EU campaigns were ongoing. Louise Edwards, the head of regulations, said both investigations were progressing very well and would be completed “definitely this year and hopefully by this summer”.

Bassett said long-established rules that every printed political communication should specify who paid for it should be extended to digital media, a practice that Facebook has said it will voluntarily adopt around the world.

She said the commission had no power to compel social media companies to hand over evidence outside the regulated period before any election, which she said “creates a real challenge”.