I lose Internet connection each time I connect to #FortiClient IPsec VPN

Check your routing table or use traceroute 1.1.1.1 to see if it is going through the tunnel. If so then you may need to enable split tunnel.

https://community.fortinet.com/t5/FortiGate/Technical-Tip-Enable-split-tunnel-For-IPsec-VPN/ta-p/192266

Do you have any firewall rules allowing traffic from your VPN to your network ? Is it NAT'ed ? If not, you would need the reverse rule to allow traffic back into your vpn.

Do you have split tunnel enabled? If so, what are you using for accessible networks?
If you don't have split tunnel enabled, do you have a policy allowing the VPN interface to the internet and to other resources within the network?