Is it true that abortion is practiced more frequently in countries with liberal abortion legislation?

No, we do not find more voluntary abortions in countries where the practice is legal. Europe and North America have the most liberal laws in terms of criteria for accessing abortion, and according to recent data (2010-2014), annual abortion rates per 1000 women aged 15 to 49 are indeed lowest in those parts of the world: 17 per 1000 in North America; and 29 per 1000 in Europe, with wide disparities between eastern and western European countries. It is important to compare these figures to estimated abortion rates in Africa (34 per 1000); Asia (36 per 1000); and Latin America (44 per 1000)—all continents where illegal - and therefore high-risk - abortion is widespread.

The relation between restricted rights to abortion and high abortion rates is explained, among other factors, by limited use of contraception in relatively underdeveloped countries. Moreover, abortion rates across all world regions have slightly yet significantly decreased over the recent decades, from 40 per 1000 women in 1990-1994 to 35 per 1000 in 2010-2014, as more and more countries legalize abortion. 

Source

Agnès Guillaume and Clémentine Rossier, 2018, Abortion around the world: an overview of legislation, measures, trends and consequences, Population 73 (2): 217-306